Systems and methods for identity-based encryption and related cryptographic techniques

ABSTRACT

A method and system for encrypting a first piece of information M to be sent by a sender [ 100]  to a receiver [ 110]  allows both sender and receiver to compute a secret message key using identity-based information and a bilinear map. In a one embodiment, the sender [ 100]  computes an identity-based encryption key from an identifier ID associated with the receiver [ 110] . The identifier ID may include various types of information such as the receiver&#39;s e-mail address, a receiver credential, a message identifier, or a date. The sender uses a bilinear map and the encryption key to compute a secret message key g ID   r , which is then used to encrypt a message M, producing ciphertext V to be sent from the sender [ 100]  to the receiver [ 110]  together with an element rP. An identity-based decryption key d ID  is computed by a private key generator [ 120]  based on the ID associated with the receiver and a secret master key s. After obtaining the private decryption key from the key generator [ 120] , the receiver [ 110]  uses it together with the element rP and the bilinear map to compute the secret message key g ID   r , which is then used to decrypt V and recover the original message M. According to one embodiment, the bilinear map is based on a Weil pairing or a Tate pairing defined on a subgroup of an elliptic curve. Also described are several applications of the techniques, including key revocation, credential management, and return receipt notification.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.11/431,410 filed May 9, 2006, which is a continuation of U.S. patentapplication Ser. No. 10/218,697 filed Aug. 13, 2002, now U.S. Pat. No.7,113,594, which claims the benefit of U.S. provisional application No.60/311,946, filed Aug. 13, 2001, all of which are incorporated herein byreference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The present invention was made with the support of DARPA contractF30602-99-1-0530. The U.S. Government has certain rights in theinvention.

BACKGROUND OF THE INVENTION

The field of the present invention relates generally to cryptographicsystems.

Public-key cryptographic systems allow two people to exchange privateand authenticated messages without requiring that they first have asecure communication channel for sharing private keys. One of the mostwidely used public-key cryptosystem is the RSA cryptosystem disclosed inU.S. Pat. No. 4,405,829. The RSA cryptosystem is currently deployed inmany commercial systems. It is used by web servers and browsers tosecure web traffic, it is used to ensure privacy and authenticity ofe-mail, it is used to secure remote login sessions, and it is at theheart of electronic credit-card payment systems. In short, RSA isfrequently used in applications where security of digital data is aconcern.

According to public-key cryptosystems such as the RSA cryptosystem, eachperson has a unique pair of keys: a private key that is a secret and apublic key that is widely known. This pair of keys has two importantproperties: (1) the private key cannot be deduced from knowledge of thepublic key alone, and (2) the two keys are complementary, i.e., amessage encrypted with one key of the pair can be decrypted only withthe complementary key. In these systems, both the public key and theprivate key in a pair are generated together as the output of a keygeneration algorithm that takes as input a random seed. Consequently, inthese cryptosystems, people cannot choose a desired public or privatekey, but must simply use the keys that are generated for them by a keygeneration algorithm. This has the disadvantage that others cannotencrypt messages to a person until that person generates and publishes apublic key. Another problem with this type of cryptosystem is that animpostor can publish a public key and claim that it belongs to someoneelse. To address this issue, a trusted certificate authority (CA) isused to authenticate individuals and certify to others that theindividual's public key is authentic. Unfortunately, this addscomplexity to the cryptosystem since a sender must obtain a certificatefor every receiver, and must obtain a new certificate every time anexisting certificate expires. It also requires receivers to createpublic keys, publish them, register certificates with the CA, and renewsuch certificates when they expire.

In 1984 Shamir envisioned a new type of public key encryption scheme(described in A. Shamir, “Identity-based cryptosystems and signatureschemes”, in Advances in Cryptology—Crypto '84, Lecture Notes inComputer Science, Vol. 196, Springer-Verlag, pp. 47-53, 1984). Accordingto Shamir's scheme, a person's public key consists of a publicidentifier, which may be the person's name and network address, orcombination of name and e-mail address, social security number, streetaddress, telephone number, or office address. Because the public key isthe person's pre-existing public identifier (ID) rather than a keyproduced from a random seed, this kind of public key cryptosystem iscalled an identity-based encryption (IBE) scheme. Shamir, however, didnot provide a concrete, practical IBE cryptosystem. In fact, Shamirargued that existing cryptosystems (such as RSA) could not be adapted torealize a secure IBE cryptosystem.

In the years since Shamir proposed his IBE scheme there have beenseveral attempts to realize an identity-based cryptosystem. Someproposals require that users not collude. Other proposals require theprivate key generator (PKG) to spend an impractically long time for eachprivate key generation request. Some proposals require tamper resistanthardware.

In short, there remains a need for improved cryptographic methods andsystems.

SUMMARY OF THE INVENTION

According to one embodiment of the invention, a method of encrypting afirst piece of information to be sent by a sender to a receiver uses anencryption key generated from a second piece of information. A bilinearmap and the encryption key are used to encrypt at least a portion of thefirst piece of information to be sent from the sender to the receiver.The bilinear map may be symmetric or asymmetric. The bilinear map may bebased on a Weil pairing or a Tate pairing defined on an algebraic groupderived from an elliptic curve. More generally, the bilinear map may bebased on a pairing defined on algebraic varieties.

According to one embodiment of the invention, encrypting the portion ofthe first piece of information can be completed prior to generating adecryption key corresponding to the encryption key.

According to another embodiment of the invention, the second piece ofinformation is known to the receiver prior to the generation of adecryption key corresponding to the encryption key. The second piece ofinformation may comprise a character string such as an e-mail address,name or other identifier associated with the receiver, according todifferent embodiments of the invention. The second piece of informationmay also include, according to various embodiments, an attributeassociated with the receiver or information corresponding to a time ortimes, such as a date or series of dates defining one or more timeintervals. A decryption key may be provided based on a time that arequest for the decryption key is received relative to the informationcorresponding to a time. According to other embodiments of theinvention, the second piece of information may include a messageidentifier, a credential identifier or a message subject identifier.

According to another embodiment of the invention, a message key isgenerated from the encryption key using a bilinear map, and acryptographic hash function is applied to the message key.

According to another embodiment of the invention, encrypting the portionof the first piece of information includes generating a mask from thesecond piece of information using a bilinear map. The mask is applied tothe portion of the second piece of information.

An embodiment of the invention is directed to a method of decryptingciphertext which has been encrypted by a sender using an identity-basedencryption key associated with a receiver. A decryption key derived fromthe encryption key is obtained. At least a portion of the ciphertext isdecrypted using a bilinear map and the decryption key. The bilinear mapmay be symmetric or asymmetric. The bilinear map may be based on a Weilpairing or a Tate pairing defined on an algebraic group derived from anelliptic curve.

According to another embodiment of the invention, the ciphertext isobtained prior to creating the decryption key. According to anotherembodiment of the invention, the first piece of information is known tothe receiver prior to obtaining the ciphertext and prior to obtainingthe decryption key. The decryption key may be obtained by sending arequest to a private key generator, including information sent togetherwith the ciphertext.

An embodiment of the invention is directed to a method of generating adecryption key corresponding to an encryption key. An algebraic group, agroup action, and a master key are provided. The encryption key isgenerated based on a first piece of information. The decryption key isgenerated based on the group action, the master key and the encryptionkey. According to one embodiment of the invention, the group action iscapable of being calculated in polynomial time. According to anotheraspect of the invention, generation of the decryption key in the absenceof the master key would require greater than polynomial time.

Another embodiment of the invention is directed to a method of providingsystem parameters for a cryptographic system. Algebraic groups

₁ and

₂ having an order q are provided, together with associated groupactions. In addition, a bilinear map is provided that maps pairs ofpoints in

₁ to points in

₂. In another embodiment, a system parameter representing a member P ofG₁, and a system parameter representing a member P_(pub) of G₁ areprovided, where P_(pub) is based on the group action of a master key sapplied to P. According to other embodiments of the invention, a systemparameter representing a set of one or more hash functions H₁, H₂, H₃,or H₄ are provided. According to another embodiment of the invention, asystem parameter representing a size n of a message space is provided.

According to another embodiment of the invention, the bilinear map maybe asymmetric or symmetric. In another embodiment the bilinear map isbased on a Weil pairing or a Tate pairing defined on a portion of anelliptic curve.

According to another embodiment of the invention, the algebraic group G₁is defined by an elliptic curve defined over a field of order p and theorder q is less than the order p. According to another aspect of theinvention, the length of p is at least 1024 bits and the length of q isno greater than 160 bits.

Another embodiment of the invention is directed to a method for managingcryptographic communication including generating shares of a master key.The shares are stored in separate systems. A request from a receiver toobtain a private key is responded to in the separate systems bygenerating from the respective shares of the master key, correspondingrespective shares of the private key. The receiver constructs theprivate key from the shares of the private key, where the private keycorresponds to identifying information of the receiver.

Another embodiment of the invention is directed to a method forcommunicating between a sender and a receiver. A message to be sent fromthe sender to the receiver is encrypted, and the message is sent fromthe sender to the receiver. A request for a decryption key is receivedfrom the receiver of the message. After receiving the request for thedecryption key, information indicating that the receiver has receivedthe message is generated, and the decryption key is provided to thereceiver. According to an embodiment of the invention, a return addressof the sender is included in the message, and an acknowledgment that themessage has been received is sent to the return address. According toanother aspect of the invention, an identification of the message isincluded in an acknowledgment and the acknowledgment is sent to thesender. According to another aspect of the invention, the encryption keyis derived based on a return address of the sender.

Another embodiment of the invention is directed to a method forcommunicating between a sender and a receiver having a credential.Identifying information of the receiver is obtained. A credentialrequired for the receiver to gain a decryption key is specified, and anencryption key is derived from the identifying information of thereceiver and the credential. A message to be sent from the sender to thereceiver is encrypted using the encryption key and a bilinear map, andthe message is sent from the sender to the receiver. A request for adecryption key is received from the receiver of the message. It isdetermined whether the receiver has the credential, and if the receiverhas the credential, the decryption key is provided to the receiver. Thereceiver then may use the decryption key and the bilinear map to decryptthe message.

Another embodiment of the invention is directed to a method ofcommunicating between a sender and a receiver involving storing adecryption key on a target system. Sets of decryption keys associatedwith times messages may be decrypted are derived, and the decryptionkeys are stored on the target system. An encryption key is derived froma string associated with a time a message is to be decrypted. A messageis encrypted using the encryption key. The message is received on thetarget system, and the message is decrypted using a bilinear map and thecorresponding decryption key.

Another embodiment of the invention is directed to a method ofcommunicating between a sender and receiver involving entities havingdifferent responsibilities. A set of decryption keys is derived from amaster key and a set of strings associated with differentresponsibilities. The decryption keys are provided to entities havingthe respective responsibilities. An encryption key is derived from astring associated with one of the different responsibilities. A messageto be sent from the sender to the receiver is encrypted using theencryption key and a bilinear map. An entity having a particularresponsibility receives the message and decrypts the message using therespective decryption key and the bilinear map. According to oneembodiment of the invention, the string corresponding to the particularresponsibility comprises a subject line of an e-mail.

BRIEF DESCRIPTION OF THE DRAWING FIGURES

FIG. 1 is a block diagram illustrating a cryptosystem according to anembodiment of the invention, showing steps taken by a sender, areceiver, and a private key generator (PKG), and informationcommunicated between them.

FIG. 2 is a block diagram illustrating steps performed by a PKG whengenerating a private key according to an embodiment of the invention.

FIG. 3 is a block diagram illustrating steps performed by a sender whencomputing a secret message key and using it to encrypt a messageintended for a receiver according to an embodiment of the invention.

FIG. 4 is a block diagram illustrating steps performed by a receiverwhen computing a secret message key and using it to decrypt ciphertextreceived from a sender according to an embodiment of the invention.

FIG. 5 is a block diagram illustrating a distributed PKG, according toan embodiment of the invention.

FIG. 6 is a block diagram illustrating elements in a cryptosystem withescrow decryption capability according to an embodiment of theinvention.

FIG. 7 is a block diagram illustrating steps performed by a sender whenencrypting messages in an ElGamal cryptosystem with escrow decryptioncapability according to an embodiment of the invention.

FIG. 8 is a block diagram illustrating steps performed by a receiverwhen decrypting messages in an ElGamal cryptosystem with escrowdecryption capability according to an embodiment of the invention.

FIG. 9 is a block diagram illustrating steps performed by an escrow whendecrypting messages in an ElGamal cryptosystem with escrow decryptioncapability according to an alternate embodiment of the invention.

FIG. 10 is a block diagram illustrating a system for managingcredentials in an identity based encryption system according to anembodiment of the invention.

FIG. 11 is a block diagram illustrating a system with key delegationaccording to an embodiment of the invention.

FIG. 12 is a block diagram illustrating an encryption system with returnreceipt according to an embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

The following description provides details of several exemplaryembodiments of the cryptographic techniques of the present invention, aswell as a technical discussion of the security of the system.

Overview

As is normally the case with modern cryptosystems, the techniques of thepresent invention are generally implemented on computers connected by acommunication medium. Although typically the computers are connected bythe Internet or another computer network, any communication medium maybe used.

One embodiment of the invention comprises an identity-based encryptionsystem that uses a secret message key derived from identity-basedinformation. The message key may be used by a sender to encrypt amessage, and by a receiver to decrypt the message. The secret messagekey is computed by the sender from an identity-based public key of thereceiver. The same message key may be computed by the receiver from thereceiver's private key, which is derived from the receiver'sidentity-based public key. Both sender and receiver compute the samesecret key using a bilinear map. For example, in one embodiment, anasymmetric or symmetric bilinear map ê:

₀×

₁→

₂ is used where

₀,

₁,

₂ are (not necessarily distinct) algebraic groups. In the case where

₀ is equal to

₁, we say the bilinear map is symmetric and often denote it as ê:

₁×

₁→

₂. A bilinear map ê that is non-degenerate and efficiently computablewill be referred to as an admissible map. It is preferable in someembodiments of the invention that the bilinear map be admissible.

The convention throughout this description will be to denote the groupoperations of

₀ and

₁ by addition, and the group operation of

₂ by multiplication. For a group

of prime order we use

* to denote the set

*=

\{O} where O is the identity element in the group

. The set of binary strings of arbitrary length is denoted by {0, 1}*.We use

_(q) to denote the group {0, . . . , q−1} under addition modulo q, andwe use

⁺ to denote the set of positive integers. We note that there is anatural group action of

_(q) on

given by repeated addition, and we denote the result of the action of anelement a ε

_(q) on an element P ε

by aP.

According to another embodiment of the invention, a certain variant(involving the map ê) of the computational Diffie-Hellman problem ishard. In one implementation the map ê is admissible and the orders of

₀,

₁,

₂ have a very large prime factor q. The orders of

₀,

₁ and

₂ may be equal to each other. Without loss of generality, the followingdescription assumes for simplicity that the orders of

₀,

₁ and

₂ are all of prime order q.

In an exemplary embodiment, an admissible map ê:

₁×

₁

₂ is used to realize an identity-based cryptosystem, as follows. Toencrypt a message, a sender uses a public key Q_(ID) ε

₁ associated with a public identifier ID for the intended receiver. Todecrypt the encrypted message, the receiver uses a complementary privatekey d_(ID) ε

₁. The private key is computed from the public key Q_(ID), a secretmaster key s ε

*_(q), and a group action of

*_(q) on

₁. In one embodiment, for example, d_(ID)=sQ_(ID). Since the secretmaster key s is known only by a trusted PKG, users normally cannotthemselves compute private keys. To obtain a private key, an individualmay obtain it from the PKG, preferably after being authenticated. At anytime, however, anyone can compute the public key Q_(ID) associated withany public identifier ID even before the corresponding private key hasbeen determined. For example, in one embodiment the public key Q_(ID)may be obtained by (1) using a conventional character encoding scheme tomap the public identifier ID to a corresponding binary string in {0,1}*, and (2) using a hash function H₁: {0, 1}*→

*₁ to hash the binary string to the element Q_(ID) of

*₁, where the order of Q_(ID) is q.

In this embodiment, a message intended for a receiver with publicidentifier ID may be encrypted and decrypted as follows. The admissiblemap ê may be used by the sender to determine a secret message key.Although the sender and receiver do not share all the same information,using the fact that the map ê is bilinear, they can use differentinformation to compute the same message key. Since each uses informationthat is private, the message key is a secret.

To illustrate how this approach may be implemented, suppose that thesender has knowledge of elements P and sP in

₁. In one embodiment, for example, elements P and P_(pub)=sP in

₁ are published system parameters. Now further suppose the senderprivately selects a random r ε

*, and uses the receiver's identity-based public key Q_(ID) to computeg_(ID) ^(r)=ê(rQ_(ID), sP). The element g_(ID) ^(r) is an identity-basedsecret which the sender may use as a secret message key to performidentity-based encryption of a message to the receiver. The sender maythen send an encrypted message together with rP to the receiver. Thereceiver then receives rP and uses it together with the private keysQ_(ID) to compute the secret message key g_(ID) ^(r)=ê(sQ_(ID), rP).This secret message key is equal to the secret message key computed bythe sender because of the bilinearity of the ê map. This computedelement g_(ID) ^(r) ε

₂ is thus an identity-based secret of the sender which the receiver maycompute using the element rP and the private key sQ_(ID). This secretmay be used as a message key for cryptographic communication between thesender and receiver.

Note that the PKG also knows the receiver's private key, so can alsocompute the secret message key and decrypt the message. The sender,receiver and PKG all have sufficient information to compute the secretmessage key. No other entity, however, normally has knowledge of thesender's secret r or the receiver's secret sQ_(ID). The security of thisembodiment is related to the difficulty of computing the secret messagekey, which is based upon a combination of r, s, and Q_(ID) using abilinear map, without knowledge of r or knowledge of sQ_(ID).

In one embodiment, the message key g_(ID) ^(r) is used to determine amask which is used to encrypt and decrypt the bits of the message usingan XOR operation (denoted by ‘⊕’). Specifically, the ciphertext V of amessage M is produced by computing V=M⊕H₂(g_(ID) ^(r)), where H₂:

₂→{0, 1}^(n) is a hash function, and n is the bit length of the message.Conversely, the message M is recovered from the ciphertext V bycomputing M=V⊕H₂(g_(ID) ^(r)).

In another embodiment, the one-way encryption scheme outlined above ismade more secure by converting it into a chosen ciphertext securesystem. In one embodiment of the invention, for example, a generaltechnique of Fujisaki-Okamoto is used.

In another embodiment, the master key is broken into components s,distributed among several private key generators in a distributed PKG.For a given user with a public key Q_(ID) based on an identifier ID,each of these private key generators in the distributed PKG computes aprivate key portion d_(i) using Q and its portion s_(i) of the masterkey. These private key portions can be combined by the user and used asa single private key d_(ID) to decrypt messages encrypted with Q_(ID).

In another embodiment, an ElGamal encryption scheme is provided withbuilt-in key escrow, i.e., where one global escrow key can decryptciphertexts encrypted under any public key. In this embodiment, theexemplary system described above is adapted as follows. Suppose that thereceiver also has knowledge of elements P and sP. Rather than obtaininga private key from the PKG, the receiver generates a public/private keypair by selecting a random x ε

*_(q), computing xP using a group action, and publishing a public keybased on the result of the computation. In one embodiment, the publickey is xP and the complementary private key is d=x(sP). (Thus, xP playsthe role of Q_(ID), and d=x(sP)=s(xP) plays the role of d_(ID)=sQ_(ID).)To encrypt a message to the receiver, the sender as before selects arandom r and sends rP to the receiver. Then the receiver knows the pair(rP, x(sP)), where x(sP)=d is a secret, while the sender knows the pair(sP, r(xP)), where r(xP) is a secret. Thus, the sender and receiver bothcan compute g=ê(rP, x(sP))=ê(sP, r(xP)), where the second equalityfollows from the bilinearity of ê. This secret, however, can also bedetermined from knowledge of the master key s. Using the element rP fromthe sender, the receiver's public key xP, and the master key s, themessage key can be computed by evaluating g=ê(rP, s(xP)). It should benoted that this embodiment makes use of a symmetric bilinear map ê:

₁×

₁→

₂.

In several embodiments of the invention,

₁ is a subgroup of an elliptic curve, and an admissible map ê isconstructed from the Weil pairing (or Tate pairing) on the ellipticcurve. (Recall that, by definition, a subgroup is not necessarilysmaller than the group, i.e.,

₁ may be the entire elliptic curve). More generally,

₁ may be an abelian variety and ê an admissible pairing of its elements.In embodiments using a map ê:

₀×

₁→

₂ where

₀ and

₁ are distinct,

₀ also may be a subgroup of an elliptic curve, or more generally, anabelian variety.

In other embodiments, various novel applications of identity-basedencryption are provided. New and useful applications of IBE systems arepossible by using other types of public identifiers, or enhanced publicidentifiers. For example, the public identifier ID is not limited to anidentifier associated with an individual person, but may be anidentifier associated with any type of entity including not justindividuals but also organizations, governmental agencies, corporationsand the like. It should also be noted that individual identities forminga group may be naturally combined to produce a joint identity for thegroup with a corresponding group private key. The group's private keyneed not be issued by a PKG, but is simply the combination of theseparate private keys of the entities composing the group. It should benoted that the basic ID specifying the identity of an entity is notlimited to the name, e-mail address, address, or social security numberof an entity, but could also include other types of information such asdomain names, URLs, 9-digit zip codes, tax identification numbers, andso on. In many applications, the public identifier ID will contain somecharacter string known to the public to be uniquely associated with aparticular entity or collection of entities. In general, however, thepublic identifier ID can be any arbitrary character string or otherarbitrary information.

Various useful applications of IBE make use of enhanced publicidentifiers. An enhanced identifier may comprise a type of identifierthat contains information not necessarily limited to informationspecifying the identity of a particular entity. For example, an ID cancontain a credential descriptor such as a license number, officialtitle, or security clearance associated with an entity. An agency canthen manage the credentials by providing private keys only to entitiesit certifies. In one exemplary embodiment, an ID can contain a propertydescriptor such as a serial number, vehicle identification number,patent number, or the like. An agency responsible for registeringproperty owners and authenticating owners can manage propertyregistration by providing private keys only to entities that itregisters as true owners. More generally, an association between two ormore things can be managed by including identifiers for them in an ID.The PKG then acts as the management authority for the associationsbetween things.

Another type of enhanced ID is an identifier that includes a time, atime interval, or a set of time intervals. A private key for such anidentifier can then be constructed to automatically expire at a certaintime, to automatically activate only after a certain time, or to bevalid only for one or more specified time intervals. This technique canbe combined with the credential and ownership management to control thetime of activation and/or expiration.

From the above examples, it is evident that an identity-based encryptionsystems according to the present invention are not limited to anyparticular type of identifier. Thus, the term ‘identity-based’ should beunderstood in general terms as indicating that any arbitrary characterstring or other arbitrary information may be used as a basis.

According to another embodiment, an IBE system allows the delegation ofdecryption capabilities. An entity can set up its own IBE system withits own secret master key, and assume the role of PKG for this IBEsystem. Because the entity has the master key, it can issue keys todelegate decryption capabilities to others. For example, if the entityis a corporation, the employees can obtain private keys from thecorporate PKG. Individuals can be issued private keys matching theirnames, titles, duties, projects, cases, or any other task-relatedidentifier. In another example, an individual can issue to a laptopprivate keys that are valid only for the duration of a business trip. Ifthe laptop is lost or stolen, only the keys for that time period arecompromised. The master key, which remained at home, is uncompromised.

It should also be pointed out that the medium of communication need notbe limited to e-mail or the Internet, but could include anycommunication medium such as printed publications, digital storagemedia, radio broadcasting, wireless communications, and so on.

DEFINITIONS

-   Identity-Based Encryption. An exemplary embodiment of an    identity-based encryption system and method ε uses four randomized    algorithms: Setup, Extract, Encrypt, Decrypt:-   Setup: Given a security parameter k, return params (system    parameters) and master-key. The system parameters include a    description of a finite message space    , and a description of a finite ciphertext space    . Normally, the system parameters will be publicly known, while the    master-key will be known only to a Private Key Generator (PKG).-   Extract: takes as input params, master-key, and an arbitrary ID ε    {0, 1}*, and returns a private key d. Here ID is an arbitrary string    that will be used as a public key, and d is the corresponding    private decryption key. The Extract algorithm extracts a private key    from the given public key. Because the extraction requires the    master-key, it is normally performed by the PKG.-   Encrypt: takes as input params, ID, and M ε    . It returns a ciphertext C ε    .-   Decrypt: takes as input params, C ε    , and a private key d. It return M ε    .    According to an embodiment of the invention, these algorithms    satisfy the standard consistency constraint that ensures decryption    will faithfully recover any encrypted message. More specifically,    when d is the private key generated by algorithm Extract when it is    given ID as the public key, then

∀Mε

: Decrypt(params, C, d)=M where C=Encrypt(params, ID, M).

In an identity-based cryptosystem according to an embodiment of theinvention, the above algorithms are used together as illustrated inFIG. 1. A sender 100 uses Encrypt, a receiver 110 uses Decrypt, and aPKG 120 uses Setup and Extract. To send a message M to receiver 110, thesender 100 obtains an ID of the receiver (e.g., the receiver's e-mailaddress) and combines it with a randomly selected integer r to compute asecret message key g_(ID) ^(r)). The element rP is sent to receiver 110who combines it with a private key d_(ID) to determine the same messagekey g_(ID) ^(r). Because the sender and receiver share the secretmessage key, a message encrypted with the key by the sender can bedecrypted by the receiver. In particular, the sender encrypts M with themessage key to produce ciphertext V which is communicated with rP to thereceiver. The receiver then uses the secret message key to decrypt theciphertext to recover the original message. In order to decryptmessages, however, the receiver 110 must first obtain the private keyd_(ID) from the PKG 120. After the PKG authenticates the identity of thereceiver, it provides the receiver with the private key corresponding tothe receiver's ID. (Note that, in this embodiment, the PKG can computeany private key in the system, and can thus decrypt any message to anyuser in the system.)

Chosen ciphertext security. Chosen ciphertext security (IND-CCA) is thestandard acceptable notion of security for a public key encryptionscheme. An embodiment of an identity-based encryption system and methodmay be implemented to satisfy this strong notion of security.Additionally, the selected level of chosen ciphertext security may bestrengthened a bit. The reason is that when an adversary attacks apublic key ID in an identity-based system, the adversary might alreadypossess the private keys of users ID₁, . . . , ID_(n) of her choice. Inan embodiment of the invention, the system remains secure under such anattack. That is, the system remains secure even when the adversary canobtain the private key associated with any identity ID_(i) of her choice(other than the public key ID being attacked). We refer to such queriesas private key extraction queries. The system of this embodiment alsoremains secure even though the adversary is challenged on a public keyID of her choice (as opposed to a random public key).

We say that an embodiment of an identity-based encryption system ormethod ε is semantically secure against an adaptive chosen ciphertextattack (IND-ID-CCA) if no polynomially bounded adversary

has a non-negligible advantage against the Challenger in the followingIND-ID-CCA game:

-   -   Setup: The challenger takes a security parameter k and runs the        Setup algorithm. It gives the adversary the resulting system        parameters params. It keeps the master-key to itself.    -   Phase 1: The adversary issues queries, q₁, . . . q_(m) where        query q_(i) is one of:        -   Extraction query            ID_(i)            . The challenger responds by running algorithm Extract to            generate the private key d_(i) corresponding to the public            key            ID_(i)            . It sends d_(i) to the adversary.        -   Decryption query            ID_(i), C_(i)            . The challenger responds by running algorithm Extract to            generate the private key d_(i) corresponding to ID_(i). It            then runs algorithm Decrypt to decrypt the ciphertext C_(i)            using the private key d_(i). It sends the resulting            plain-text to the adversary.    -   These queries may be asked adaptively, that is, each query q,        may depend on the replies to, q₁, . . . , q_(i−1).    -   Challenge: Once the adversary decides that Phase 1 is over it        outputs two equal length plain-texts M₀, M₁ ε        and an identity ID on which it wishes to be challenged. The only        constraint is that ID did not appear in any private key        extraction query in Phase 1.        -   The challenger picks a random bit b ε {0, 1} and sets            C=Encrypt(params, ID, M_(b)). It sends C as the challenge to            the adversary.    -   Phase 2: The adversary issues more queries, q_(m+1), . . . ,        q_(n) where query q_(i) is one of:        -   Extraction query            ID_(i)            where ID_(i)≠ID. Challenger responds as in Phase 1.        -   Decryption query            ID_(i), C_(i)            ≠            ID, C            . Challenger responds as in Phase 1.    -   These queries may be asked adaptively as in Phase 1.    -   Guess: Finally, the adversary outputs a guess b′ ε {0, 1}. The        adversary wins the game if b=b′.    -   We refer to such an adversary        as an IND-ID-CCA adversary. We define adversary        's advantage in attacking the scheme ε as the following function        of the security parameter k (k is given as input to the        challenger):

${(k)} = {{{{\Pr \left\lbrack {b = b^{\prime}} \right\rbrack} - \frac{1}{2}}}.}$

-   -   The probability is over the random bits used by the challenger        and the adversary.        Using the IND-ID-CCA game we can define chosen ciphertext        security for IBE schemes. As usual, we say that a function g:        →        is negligible if g(k) is smaller than 1/ƒ(k) for any polynomial        ƒ.        Definition 1 We say that an IBE system ε is semantically secure        against an adaptive chosen ciphertext attack if for any        polynomial time IND-ID-CCA adversary        the function        (k) is negligible. As shorthand, we say that ε is IND-ID-CCA        secure.

Note that the standard definition of chosen ciphertext security(IND-CCA) is the same as above except that there are no private keyextraction queries and the adversary is challenged on a random publickey (rather than a public key of her choice). Private key extractionqueries are related to the definition of chosen ciphertext security inthe multiuser settings. After all, our definition involves multiplepublic keys belonging to multiple users. A multiuser IND-CCA may bereducible to single user IND-CCA using a standard hybrid argument. Thisdoes not hold in the identity-based settings, IND-ID-CCA, since theadversary gets to choose which public keys to corrupt during the attack.To emphasize the importance of private key extraction queries we notethat one implementation of the disclosed IBE system can be modified (byremoving one of the hash functions) into a system which has chosenciphertext security when private extraction queries are disallowed.However, the implementation is insecure when extraction queries areallowed.

Semantically secure identity based encryption. The proof of security foran implementation of our IBE system makes use of a weaker notion ofsecurity known as semantic security (also known as semantic securityagainst a chosen plain-text attack). Semantic security is similar tochosen ciphertext security (IND-ID-CCA) except that the adversary ismore limited; it cannot issue decryption queries while attacking thechallenge public key. For a standard public key system (not an identitybased system) semantic security is defined using the following game: (1)the adversary is given a random public key generated by the challenger,(2) the adversary outputs two equal length messages M₀ and M₁ andreceives the encryption of M_(b) from the challenger where b is chosenat random in {0, 1}, (3) the adversary outputs b′ and wins the game ifb=b′. The public key system is said to be semantically secure if nopolynomial time adversary can win the game with a non-negligibleadvantage. As shorthand we say that a semantically secure public keysystem is IND-CPA secure. Semantic security captures our intuition thatgiven a ciphertext the adversary learns nothing about the correspondingplain-text.

To define semantic security for identity based systems (denotedIND-ID-CPA) we strengthen the standard definition by allowing theadversary to issue chosen private key extraction queries. Similarly, theadversary is challenged on a public key ID of her choice. We definesemantic security for identity based encryption schemes using anIND-ID-CPA game. The game is identical to the IND-ID-CCA game definedabove except that the adversary cannot make any decryption queries. Theadversary can only make private key extraction queries. We say that anidentity-based encryption scheme ε is semantically secure (IND-ID-CPA)if no polynomially bounded adversary

has a non-negligible advantage against the Challenger in the followingIND-ID-CPA game:

-   -   Setup: The challenger takes a security parameter k and runs the        Setup algorithm. It gives the adversary the resulting system        parameters params. It keeps the master-key to itself.    -   Phase 1: The adversary issues private key extraction queries        ID₁, . . . , ID_(m). The challenger responds by running        algorithm Extract to generate the private key d_(i)        corresponding to the public key ID_(i). It sends d_(i) to the        adversary. These queries may be asked adaptively.    -   Challenge: Once the adversary decides that Phase 1 is over it        outputs two equal length plain-texts M₀, M₁ ε        and a public key ID on which it wishes to be challenged. The        only constraint is that ID did not appear in any private key        extraction query in Phase 1. The challenger picks a random bit b        ε {0, 1} and sets C=Encrypt(params, ID, M_(b)). It sends C as        the challenge to the adversary.    -   Phase 2: The adversary issues more extraction queries ID_(m+1),        . . . , ID_(n). The only constraint is that ID_(i)≠ID. The        challenger responds as in Phase 1.    -   Guess: Finally, the adversary outputs a guess b′ ε {0, 1}. The        adversary wins the game if b=b′.    -   We refer to such an adversary        as an IND-ID-CPA adversary. As we did above, the advantage of an        IND-ID-CPA adversary        against the scheme E is the following function of the security        parameter k:

${(k)} = {{{{\Pr \left\lbrack {b = b^{\; \prime}} \right\rbrack} - \frac{1}{\; 2}}}.}$

-   -   The probability is over the random bits used by the challenger        and the adversary.        Definition 2 We say that the IBE system ε is semantically secure        if for any polynomial time IND-ID-CPA adversary        the function        (k) is negligible. As shorthand, we say that ε is IND-ID-CPA        secure.        One way identity-based encryption. One can define an even weaker        notion of security called one-way encryption (OWE). Roughly        speaking, a public key encryption scheme is a one-way encryption        if given the encryption of a random plain-text the adversary        cannot produce the plain-text in its entirety. One-way        encryption is a weak notion of security since there is nothing        preventing the adversary from, say, learning half the bits of        the plain-text. Hence, one-way encryption schemes do not        generally provide secure encryption. In the random oracle model        one-way encryption schemes can be used for encrypting        session-keys (the session-key is taken to be the hash of the        plain-text). We note that one can extend the notion of one-way        encryption to identity based systems by adding private key        extraction queries to the definition. We do not give the full        definition here since we use semantic security as the weakest        notion of security.

Bilinear Maps and the Bilinear Diffie-Hellman Assumption

One embodiment of the invention is directed to an IBE system that makesuse of a map ê:

₁×

₁→

₂ between groups

₁ and

₂ of order q for some large prime q. A map ê may be called an admissiblemap if it satisfies the following properties:

-   1. Bilinear: The map ê:    ₁×    ₁→    ₂ satisfies ê(aP, bQ)=ê(P, Q)^(ab) for all (P, Q) ε    ₁×    ₁ and all a, b ε    .-   2. Non-degenerate: The map does not send all pairs in    ₁×    ₁ to the identity in    ₂. Observe that since    ₁,    ₁,    ₂ are groups of prime order this implies that if    ₁=    ₁ and P is a generator of    ₁=    ₁ then ê(P, P) is a generator of    ₂.-   3. Computable: There is an efficient algorithm to compute ê(P, Q)    for any (P, Q) ε    ₁×    ₁.    Although many of the embodiments are described with reference to a    map ê:    ₁×    ₁→    ₂, this is only a specific case of bilinear maps used in embodiments    of the invention. More generally, maps ê:    ₀×    ₁→    ₂ may be used, where    ₀ and    ₁ may be distinct. For simplicity of description, however, the    description of many of the embodiments focuses primarily on the case    where    ₁ and    ₁ are the same, and both groups are then denoted    ₁. Below we present a detailed exemplary embodiment using groups    ₁,    ₂ and an admissible map between them. In this exemplary embodiment,    the group    ₁ is a subgroup of the additive group of points of an elliptic curve    E/    _(p), and the group    ₂ is a subgroup of the multiplicative group of a finite field    *_(p) ₂ . As we will see below in the detailed example of an IBE    system, the Weil pairing (which is not itself an admissible map) can    be used to construct an admissible map between these two groups.

The existence of the admissible map ê:

₁×

₁→

₂ as above has two direct implications to these groups.

-   The MOV reduction: The discrete log problem in    ₁ is no harder than the discrete log problem in    ₂. To see this, let P, Q ε    ₁ be an instance of the discrete log problem in    ₁ where both P, Q have order q. We wish to find an α ε    _(q) such that Q=αP. Let g=ê(P, P) and h=ê(Q, P). Then, by    bilinearity of êwe know that h=g^(α). By non-degeneracy of ê both g,    h have order q in    ₂. Hence, we reduced the discrete log problem in    ₁ to a discrete log problem in    ₂. It follows that for discrete log to be hard in    ₁ we must choose our security parameter so that discrete log is hard    in    ₂.-   Decision Diffie-Hellman is Easy: The Decision Diffie-Hellman problem    (DDH) in    ₁ is the problem of distinguishing between the distributions    P, aP, bP, abP    and    P, aP, bP, cP    where a, b, c are random in    _(g), and P is random in    ₁. To see that DDH in    ₁ is easy, observe that given P, aP, bP, cP ε    *₁ we have

c=ab mod q

ê(P, cP)={circumflex over (e)}(aP, bP).

-   -   The Computational Diffie-Hellman problem (CDH) in        ₁ can still be hard (CDH in        ₁ is to find abP given random        P, aP, bP        ). Exemplary embodiments may use mappings ê:        ₁×        ₁→        ₂ where CDH in        ₁ is believed to be hard even though DDH in        ₁ is easy.

The Bilinear Diffie-Hellman Assumption (BDH)

Since the Decision Diffie-Hellman problem (DDH) in

₁ is easy, embodiments of the invention do not use DDH to buildcryptosystems in the group

₁. Instead, the security in embodiments of our IBE system is based on anovel variant of the Computational Diffie-Hellman assumption called theBilinear Diffie-Hellman Assumption (BDH).

Bilinear Diffie-Hellman Problem. Let

₁,

₂ be two groups of prime order q. Let ê:

₁×

₁→

₂ be an admissible map and let P be a generator of

₁. The BDH problem in

₁,

₂, ê

is as follows: Given

P, aP, bP, cP

for some a, b, c ε

_(q)* compute W=ê(P, P)^(abc) ε

₂. A algorithm

has advantage ε in solving BDH in

₁,

₂, ê

if

Pr[

(P, aP, bP, cP)={circumflex over (e)}(P, P)^(abc)]≧ε

where the probability is over the random choice of a, b, c in

*_(q), the random choice of P ε

*₁, and the random bits of

.BDH Parameter Generator. We say that a randomized algorithm

is a BDH parameter generator if (1)

takes a security parameter k ε

⁺, (2)

runs in polynomial time in k, and (3)

outputs a prime number q, the description of two groups

₁,

₂ of order q, and the description of an admissible map ê:

₁×

₁→

₂. We denote the output of

by

(1^(k))=

q,

₁,

₂, ê

. The security parameter k is used to determine the size of q; forexample, one could take q to be a random k-bit prime. For i=1, 2 weassume that the description of the group

_(i) contains polynomial time (in k) algorithms for computing the groupaction in

, and contains a generator of

. The generator of

, enables us to generate uniformly random elements in

. Similarly, we assume that the description of ê contains a polynomialtime algorithm for computing ê. We give an example of a BDH parametergenerator below in the detailed example of an IBE system using the Weilpairing.Bilinear Diffie-Hellman Assumption. Let

be a BDH parameter generator. We say that an algorithm

has advantage ε(k) in solving the BDH problem for

if for sufficiently large k:

${(k)} = {{\Pr \left\lbrack {{\left( {q,_{1},_{2},\hat{e},P,{aP},{bP},{cP}} \right)} = {{{\hat{e}\left( {P,P} \right)}^{abc}}\begin{matrix}{\left. {\langle{q,_{1},_{2},\hat{e}}\rangle}\leftarrow{\left( 1^{k} \right)} \right.,} \\{\left. P\leftarrow _{1}^{*} \right.,a,b,\left. c\leftarrow{\mathbb{Z}}_{q}^{*} \right.}\end{matrix}}} \right\rbrack} > {\varepsilon (k)}}$

We say that

satisfies the BDH assumption if for any randomized polynomial time (ink) algorithm

and for any polynomial ƒ ε

[x] algorithm

solves the BDH problem with advantage at most 1/ƒ(k). When

satisfies the BDH assumption we say that BDH is hard in groups generatedby

.

In the description below of a detailed example of an IBE system we givesome examples of BDH parameter generators that are believed to satisfythe BDH assumption.

Hardness of BDH. It is interesting to study the relationship of the BDHproblem to other hard problems used in cryptography. Currently, all wecan say is that the BDH problem in

₁,

₂, ê

is no harder than the CDH problem in

₁ or

₂. In other words, an algorithm for CDH in

₁ or

₂ is sufficient for solving BDH in

₁,

₂, ê

. The converse is currently an open problem: is an algorithm for BDHsufficient for solving CDH in

₁ or in

₂?

We note that in a detailed example of an IBE system below, theisomorphisms from

₁ to

₂ induced by the admissible map are believed to be one-way functions.More specifically, for a point Q ε

*₁ define the isomorphism f_(Q):

₁→

₂ by f_(Q)(P)=ê(P, Q). If any one of these isomorphisms turns out to beinvertible, then BDH is easy in

₁,

₂, ê

. Fortunately, an efficient algorithm for inverting f_(Q) would imply anefficient algorithm for deciding DDH in the group

₂. In the exemplary embodiments DDH is believed to be hard in the group

₂. Hence, the isomorphisms f_(Q):

₁→

₂ induced by the admissible map are believed to be one-way functions.

Exemplary Identity-Based Encryption Scheme

We describe the following exemplary embodiments in stages. First wedescribe a basic identity-based encryption system and method which isnot secure against an adaptive chosen ciphertext attack. Anotherembodiment described below extends the basic scheme to get securityagainst an adaptive chosen ciphertext attack (IND-ID-CCA) in the randomoracle model. We later relax some of the requirements on the hashfunctions to provide alternative embodiments. These embodiments aredescribed with reference to a generic BDH parameter generator gsatisfying the BDH assumption. Later we describe a detailed example ofan IBE system using the Weil pairing.

BasicIdent

The following describes a basic embodiment, called BasicIdent. Wepresent the embodiment by describing the four algorithms: Setup,Extract, Encrypt, Decrypt. We let k be the security parameter given tothe setup algorithm. We let

be some BDH parameter generator.

-   Setup: Given a security parameter k ε    ⁺, the algorithm in the basic embodiment works as follows:    -   Step 1: Run        on input k to generate a prime q, two groups        ₁,        ₂ of order q, and an admissible map ê:        ₁×        ₁→        ₂. Choose an arbitrary generator P ε        ₁.    -   Step 2: Pick a random s ε        *_(q) and set P_(pub)=sP.    -   Step 3: Choose a cryptographic hash function H₁: {0, 1}*→        ₁*. Choose a cryptographic hash function H₂:        ₂→{0, 1}^(n) for some n. The security analysis will view H₁, H₂        as random oracles.    -   The message space is        ={0, 1}^(n). The ciphertext space is        =        *₁×{0, 1}^(n). The system parameters are params=        q,        ₁,        ₂, ê, n, P, P_(pub), H₁, H₂        . The master-key is s ε        *_(q). Embodiments of the IBE system may be used to encrypt a        symmetric key, in which case one may take n to be, for example,        128 or 256. For k one may use, for example, 512 or 1024 or 2048.-   Extract: For a given string ID ε {0, 1}* the algorithm in the basic    embodiment does: (1) computes Q_(ID)=H₁(ID) ε    *₁, and (2) sets the private key d_(ID) to be d_(ID)=sQ_(ID) where s    is the master key.    -   Extract may be performed by a PKG in various embodiments as        illustrated in FIG. 2. The PKG obtains the master key in block        200, obtains the public identifier ID in block 210, computes the        public key from the ID in block 220, then computes the private        key from the master key and the public key in block 230. In        block 240 the private key is then sent to an entity associated        with the public identifier ID, normally after the entity's        identity has been authenticated.-   Encrypt: To encrypt M ε    under the public key ID do the following: (1) compute Q_(ID)=H₁(ID)    ε    *₁, (2) choose a random r ε    *_(q), and (3) set the ciphertext to be

C=

rP, M⊕H ₂(g _(ID) ^(r))

where g _(ID) =ê(Q _(ID) ,P _(pub)) ε

*₂.

-   -   In the basic embodiment, the sender of a message may perform        Encrypt as illustrated in FIG. 3. In block 300 the system        parameters are obtained (from an external resource such as a        PKG, or from a local storage medium if they were obtained        previously). A receiver's ID is determined in block 310, and the        corresponding public key is computed from the ID in block 320.        Then the secret message key is computed in block 330, and the        message key is used to encrypt the message in block 340.

-   Decrypt: Let C=    U, V    ε    be a ciphertext encrypted using the public key ID. To decrypt C    using the private key d_(ID) ε    *₁ compute:

V⊕H ₂({circumflex over (e)}(d _(ID) ,U))=M.

-   -   In the basic embodiment, the receiver may perform Decrypt as        illustrated in FIG. 4. In block 400, the system parameters are        obtained (from an external resource such as a PKG, or from a        local storage medium if they were obtained previously). The        ciphertext V and an element rP are obtained from the sender in        block 410. The element rP may be considered a portion of the        total ciphertext obtained from the sender. In block 420 the        receiver obtains the private key d_(ID) corresponding to the        public identifier ID used to encrypt the message. The private        key is normally obtained from an external resource such as a        PKG, or from a local storage medium if it was obtained        previously. The secret message key is then computed in block        430, and used to decrypt the message in block 440.        This completes the description of BasicIdent for the basic        embodiment. We first verify consistency. When everything is        computed as above we have:        1. During encryption M is bitwise exclusive-ored with the hash        of: g_(ID) ^(r).        2. During decryption V is bitwise exclusive-ored with the hash        of: ê(d_(ID), U).        These masks used during encryption and decryption are the same        since:

{circumflex over (e)}(d _(ID) ,U)={circumflex over (e)}(sQ _(ID),rP)={circumflex over (e)}(Q _(ID) ,P _(pub))^(sr)={circumflex over(e)}(Q _(ID) ,P _(pub))^(r) =g _(ID) ^(r)

Thus, applying decryption after encryption produces the original messageM as required. Performance considerations of BasicIdent are discussedlater.Security. Next, we study the security of this basic embodiment.

The security of the exemplary system is based on the assumption that avariant of the Computational Diffie-Hellman problem in

₁ is hard. The technical security details of the encryption scheme arediscussed by the inventors in D. Boneh, M. Franklin, “Identity basedencryption from the Weil pairing”, extended abstract in Advances inCryptology—Crypto 2001, Lecture Notes in Computer Science, Vol. 2139,Springer-Verlag, pp. 231-229, 2001, which is incorporated herein byreference.

In an exemplary embodiment, the performance of the system is comparableto the performance of ElGamal encryption in

*_(p). The security of the exemplary system is based on a variant of thecomputational Diffie-Hellman assumption. Based on this assumption weshow that the exemplary system has chosen ciphertext security in therandom oracle model. In accordance with a distributed PKG embodiment,threshold cryptography techniques allow the PKG to be distributed sothat the master-key is never available in a single location. Unlikecommon threshold systems, we show that robustness for the distributedPKG embodiment is free.

To argue about the security of the exemplary system, we define chosenciphertext security for identity-based encryption. Our model gives theadversary more power than the standard model for chosen ciphertextsecurity. First, we allow the attacker to attack an arbitrary public keyID of her choice. Second, while mounting a chosen ciphertext attack onID we allow the attacker to obtain from the PKG the private key for anypublic key of her choice, other than the private key for ID. This modelsan attacker who obtains a number of private keys corresponding to someidentities of her choice and then tries to attack some other public keyID of her choice. Even with the help of such queries, it is desirablethat the attacker still have negligible advantage in defeating thesemantic security of the system.

The following theorem shows that BasicIdent is a semantically secureidentity based encryption scheme (IND-ID-CPA) assuming BDH is hard ingroups generated by

.

Theorem 1 Suppose the hash functions H₁, H₂ are random oracles. ThenBasicIdent is a semantically secure identity based encryption scheme(IND-ID-CPA) assuming BDH is hard in groups generated by

. Concretely, suppose there is an IND-ID-CPA adversary

that has advantage ε(k) against the scheme BasicIdent. Suppose

makes at most q_(E)>0 private key extraction queries and q_(H) ₂ >0 hashqueries to H₂. Then there is an algorithm

that solves BDH in groups generated by

with advantage at least:

${(k)} \geq \frac{2{\varepsilon (k)}}{{\left( {1 + q_{E}} \right)} \cdot q_{H_{2}}}$

Here e≈2.71 is the base of the natural logarithm. The running time of

is O(time(

)).

To prove the theorem we first define a related Public Key Encryptionscheme (not an identity based scheme), called BasicPub. BasicPub isdescribed by three algorithms: keygen, encrypt, decrypt.

keygen: Given a security parameter k ε

⁺, the algorithm works as follows:

-   -   Step 1: Run        on input k to generate two prime order groups        ₁,        ₂ and an admissible map ê:        ₁×        ₁→        ₂. Let q be the order of        ₁,        ₂. Choose an arbitrary generator P ε        ₁.    -   Step 2: Pick a random s ε        *_(q) and set P_(pub)=sP. Pick a random Q_(ID), ε        *₁.    -   Step 3: Choose a cryptographic hash function H₂:        ₂→{0, 1}^(n) for some n.    -   Step 4: The public key is        q,        ₁,        ₂, ê, n, P, P_(pub), Q_(ID), H₂        . The private key is d_(ID)=sQ_(ID), ε        *₁.        encrypt: To encrypt M ε {0, 1}^(n) choose a random r ε        *_(q) and set the ciphertext to be:

C=

rP,M⊕H ₂(g ^(r))

where g=ê(Q _(ID) ,P _(pub)) ε

*₂

decrypt: Let C=

U, V

be a ciphertext created using the public key

q,

₁,

₂, ê, n, P, P_(pub), Q_(ID), H₂

.

To decrypt C using the private key d_(ID) ε

*₁ compute:

V|H ₂({circumflex over (e)}(d _(ID) ,U))=M

This completes the description of BasicPub. We now prove Theorem 1 intwo steps. We first show that an IND-ID-CPA attack on BasicIdent can beconverted to a IND-CPA attack on BasicPub. This step shows that privatekey extraction queries do not help the adversary. We then show thatBasicPub is IND-CPA secure if the BDH assumption holds. The proofs areomitted.Lemma 2 Let H₁ be a random oracle from {0, 1}* to

*₁ Let

be an IND-ID-CPA adversary that has advantage ε(k) against BasicIdent.Suppose

makes at most q_(E)>0 private key extraction queries. Then there is aIND-CPA adversary

that has advantage at least ε(k)/e(1+q_(E)) against BasicPub. Itsrunning time is O(time(

)).Lemma 3 Let H₂ be a random oracle from

₂ to {0, 1}^(n). Let

be an IND-CPA adversary that has advantage ε(k) against BasicPub.Suppose

makes a total of q_(H) ₂ >0 queries to H₂. Then there is an algorithm

that solves the BDH problem for

with advantage at least 2ε(k)/q_(H) ₂ and a running time O(time(

)).Proof of Theorem 1. The theorem follows directly from Lemma 2 and Lemma3. Composing both reductions shows that an IND-ID-CPA adversary onBasicIdent with advantage ε(k) gives a BDH algorithm for

with advantage at least 2ε(k)/e(1+q_(E))q_(H) ₂ , as required. □Identity-Based Encryption with Chosen Ciphertext Security

According to one embodiment of the invention, a technique of Fujisakiand Okamoto (described in E. Fujisaki and T. Okamoto, “Secureintegration of asymmetric and symmetric encryption schemes”, in Advancesin Cryptology—Crypto '99, Lecture Notes in Computer Science, Vol. 1666,Springer-Verlag, pp. 537-554, 1999, which is incorporated herein byreference) may be appropriately adapted to convert the BasicIdentembodiment of the previous section into a chosen ciphertext secureembodiment of an IBE system (in the sense defined earlier) in the randomoracle model. Let ε be a probabilistic public key encryption scheme. Wedenote by ε_(pk)(M; r) the encryption of M using the random bits r underthe public key p k. Fujisaki-Okamoto define the hybrid scheme E^(hy) as:

E _(pk) ^(hy)(M)=

E _(pk)(σ; H ₃(σ,M)), H ₄(σ)⊕M

Here σ is generated at random and H₃, H₄ are cryptographic hashfunctions. Fujisaki-Okamoto show that if E is a one-way encryptionscheme then E^(hy) is a chosen ciphertext secure system (IND-CCA) in therandom oracle model (assuming E_(pk) satisfies some naturalconstraints). We note that semantic security implies one-way encryptionand hence the Fujisaki-Okamoto result also applies if E is semanticallysecure (IND-CPA).

We apply the Fujisaki-Okamoto transformation to BasicIdent and show thatthe resulting embodiment of an IBE system is IND-ID-CCA secure. Weobtain the following IBE embodiment which we call FullIdent. Recall thatn is the length of the message to be encrypted.

-   Setup: As in the BasicIdent scheme. In addition, we pick a hash    function H₃: {0, 1}^(n)×{0, 1}^(n)→    *_(q), and a hash function H₄: {0, 1}^(n)→{0, 1}^(n).-   Extract: As in the BasicIdent scheme.-   Encrypt: To encrypt M ε {0, 1}^(n) under the public key ID do the    following: (1) compute Q_(ID)=H₁(ID) ε    , (2) choose a random σ ε {0, 1}^(n), (3) set r=H₃(σ, M), and (4)    set the ciphertext to be

C=

rP, σ⊕H ₂(g _(ID) ^(r)), M⊕H ₄(σ)

where g _(ID) =ê(Q _(ID) ,P _(pub)) ε

₂

-   Decrypt: Let C=    U, V, W    be a ciphertext encrypted using the public key ID. If U ∉    *₁ reject the ciphertext. To decrypt C using the private key d_(ID)    ε    *₁ do:

1. Compute V⊕H₂(ê(d_(ID, U)=σ.)

2. Compute W⊕H₄(σ)=M.

3. Set r=H₃(σ, M). Test that U=rP. If not, reject the ciphertext.

4. Output M as the decryption of C.

This completes the description of FullIdent. Its implementation followsthe same basic pattern as BasicIdent shown in FIGS. 2, 3, 4. Note that Mis encrypted as W=M⊕H₄(σ). This can be replaced by W=E_(H) ₄ _((σ))(M)where E is a semantically secure symmetric encryption scheme.Security. The following theorem shows that FullIdent is a chosenciphertext secure IBE (i.e. IND-ID-CCA), assuming BDH is hard in groupsgenerated by

.Theorem 4 Let the hash functions H₁, H₂H₃, H₄ be random oracles. ThenFullIdent is a chosen ciphertext secure IBE (IND-ID-CCA) assuming BDH ishard in groups generated by

.Concretely, suppose there is an IND-ID-CCA adversary

that has advantage ε(k) against the scheme FullIdent and

runs in time at most t(k). Suppose

makes at most q, extraction queries, at most q_(D) decryption queries,and at most q_(H) ₂ , q_(H) ₃ , q_(H) ₄ queries to the hash functionsH₂, H₃, H₄ respectively. Then there is a BDH algorithm

for

with running time t₁(k) where:

${(k)} \geq {2{{{FO}_{adv}\left( {\frac{\varepsilon (k)}{\left( {1 + q_{E} + q_{D}} \right)},q_{H_{4}},q_{H_{3}},q_{D}} \right)}/q_{H_{2}}}}$t₁(k) ≤ FO_(time)(t(k), q_(H₄), q_(H₃))

where the functions FO_(time) and FO_(adv) are defined in Theorem 5.

The proof of Theorem 4 is based on the following result of Fujisaki andOkamoto. Let BasicPub^(hy) be the result of applying theFujisaki-Okamoto transformation to BasicPub.

Theorem 5 (Fujisaki-Okamoto) Suppose

is an IND-CCA adversary that achieves advantage ε(k) when attackingBasicPub^(hy). Suppose

has running time t(k), makes at most q, decryption queries, and makes atmost q_(H) ₃ , q_(H) ₄ queries to the hash functions H₃, H₄respectively. Then there is an IND-CPA adversary

against BasicPub with running time t₁(k) and advantage ε₁(k) where

${{\varepsilon_{1}(k)} \geq {{FO}_{adv}\left( {{\varepsilon (k)},q_{H_{4}},q_{H_{3}},q_{D}} \right)}} = {\frac{1}{2\left( {q_{H_{4}} + q_{H_{3}}} \right)}\left\lbrack {{\left( {{\varepsilon (k)} + 1} \right)\left( {1 - {2/q}} \right)^{q_{D}}} - 1} \right\rbrack}$t₁(k) ≤ FO_(time)(t(k), q_(H₄), q_(H₃)) = t(k) + O((q_(H₄) + q_(H₃)) ⋅ n),  and

Here q is the size of the groups

₁,

₂ and n is the length of σ.

In fact, Fujisaki-Okamoto prove a stronger result: Under the hypothesisof Theorem 5, BasicPub^(hy) would not even be a one-way encryptionscheme. For our purposes the result in Theorem 5 is sufficient. To proveTheorem 4 we also need the following lemma to trans-late between anIND-ID-CCA chosen ciphertext attack on FullIdent and an IND-CCA chosenciphertext attack on BasicPub^(hy).

Lemma 6 Let

be an IND-ID-CCA adversary that has advantage ε(k) against FullIdent.Suppose

makes at most q_(E)>0 private key extraction queries and at most q_(D)decryption queries. Then there is an IND-CCA adversary

that has advantage at least

$\frac{\varepsilon (k)}{\left( {1 + q_{E} + q_{D}} \right)}$

against BasicPub^(hy). Its running time is O(time(

)).Proof of Theorem 4. By Lemma 6 an IND-ID-CCA adversary on FullIdentimplies an IND-CCA adversary on BasicPub^(hy). By Theorem 5 an IND-CCAadversary on BasicPub^(hy) implies an IND-CPA adversary on BasicPub. ByLemma 3 an IND-CPA adversary on BasicPub implies an algorithm for BDH.Composing all these reductions gives the required bounds. □

Relaxing the Hashing Requirements

Recall that an IBE system of Section uses a hash function H₁: {0, 1}*→

. The detailed example of an IBE system presented in the next sectionuses

₁ as a subgroup of the group of points on an elliptic curve. Inpractice, it sometimes can be difficult to build hash functions thathash directly onto such groups. In an exemplary embodiment, we thereforeshow how to relax the requirement of hashing directly onto

*₁. Rather than hash onto

₁ we hash onto some set A ⊂ {0, 1}* and then use a deterministicencoding function to map A onto

*₁.

Admissible encodings: Let

₁ be a group and let A ε {0, 1}* be a finite set. We say that anencoding function L: A→

*₁ is admissible if it satisfies the following properties:

-   1. Computable: There is an efficient deterministic algorithm to    compute L(x) for any x ε A.-   2. l-to-1: For any y ε    *₁ the preimage of y under L has size exactly l. In other words,    |L⁻¹(y)|=l for all y ε    *₁. Note that this implies that |A|=l·|    *₁|.-   3. Samplable: There is an efficient randomized algorithm    _(S) such that    _(S)(y) induces a uniform distribution on L⁻¹(y) for any y ε    *₁. In other words,    _(S)(y) is a uniform random element in L⁻¹(y).    We modify FullIdent to obtain an IND-ID-CCA secure embodiment of an    IBE system where H₁ is replaced by a hash function into some set A.    Since the change is relatively minor we refer to this new scheme as    FullIdent′:-   Setup: As in the FullIdent embodiment. The only difference is that    H₁ is replaced by a hash function H′₁: {0, 1}*→A. The system    parameters also include a description of an admissible encoding    function L: A→    *₁.-   Extract, Encrypt: As in the FullIdent embodiment. The only    difference is that in Step 1 these algorithms compute    Q_(ID)=L(H′₁(ID))ε    *₁.-   Decrypt: As in the FullIdent embodiment.

This completes the description of FullIdent′. The following theoremshows that FullIdent′ is a chosen ciphertext secure IBE (i.e.IND-ID-CCA), assuming FullIdent is.

Theorem 7 Let

be an IND-ID-CCA adversary on FullIdent′ that achieves advantage ε(k).Suppose

makes at most q_(H) ₁ queries to the hash function H′₁. Then there is anIND-ID-CCA adversary

on FullIdent that achieves the same advantage ε(k) and time(

)=time(

)+q_(H) ₁ ·time(L_(S))Proof Sketch Algorithm

attacks FullIdent by running algorithm

. It relays all decryption queries, extraction queries, and hash queriesfrom

directly to the challenger and relays the challenger's response back to

. It only behaves differently when

issues a hash query to H′₁. Recall that

only has access to a hash function H₁: {0, 1}*→

*₁. To respond to H′₁ queries algorithm

maintains a list of tuples

ID_(j), y_(j)

as explained below. We refer to this list as the (H′₁)^(list). The listis initially empty. When

queries the oracle H′₁ at a point ID_(i) algorithm

responds as follows:

-   1. If the query ID, already appears on the (H′₁)^(list) in a tuple    ID_(j), y_(j)    then respond with H′₁(ID_(i))=y_(i) ε A.-   2. Otherwise,    issues a query for H₁(ID_(i)). Say, H₁(ID_(i))=α ε    *₁.-   3.    runs the sampling algorithm    _(S)(σ) to generate a random element y ε L⁻¹(α).-   4.    adds the tuple    ID_(i), y    to the (H′₁)^(list) and responds to    with H′₁(ID_(i))=y ε A. Note that y is uniformly distributed in A as    required since α is uniformly distributed in    *₁ and L is an l-to-1 map.    Algorithm    's responses to all of    's queries, including H′₁ queries, are identical to    's view in the real attack. Hence,    will have the same advantage ε(k) in winning the game with the    challenger. □

A Detailed Example of an IBE System Using the Weil Pairing

In this section we use FullIdent′ to describe a detailed example of anembodiment of an IBE system. This embodiment is based on the Weilpairing. Although in practice the Tate pairing has computationaladvantages and may be used instead of the Weil pairing in variousembodiments, the implementation using the Weil pairing will be describedfirst because it is simpler. Later, the Tate pairing will be discussed.

Properties of the Weil Pairing

Let p>3 be a prime satisfying p=2 mod 3 and let q be some prime factorof p+1. Let E be the elliptic curve defined by the equation y²=x³+1 over

_(p). We state a few elementary facts about this curve E. From here onwe let E(

_(p) _(r) ) denote the group of points on E defined over

_(p) _(r) .

-   Fact 1: Since x³+1 is a permutation on    _(p) it follows that the group E(    _(p)) contains p+1 points. We let O denote the point at infinity.    Let P ε E(    _(p)) be a point of order q and let    G₁ be the subgroup of points generated by P.-   Fact 2: For any y_(o) ε    _(p) there is a unique point (x_(o), y_(o)) on E(    _(p)), namely x_(o)=(y_(o) ²−1)^(1/3) ε    _(p). Hence, if (x, y) is a random non-zero point on E(    _(p)) then y is uniform in    _(p). We use this property to build a simple admissible encoding    function.-   Fact 3: Let 1≠ζ    _(p) ₂ be a solution of x³−1=0 mod p. Then the map φ(x, y)=(ζx, y)    is an automorphism of the group of points on the curve E. Note that    for any point Q=(x, y) ε E(    _(p)) we have that φ(Q) ε E(    _(p) ₂ ), but φ(Q) ∉ E(    _(p)). Hence, Q ε E(    _(p)) is linearly independent of φ(Q) ε E(    _(p) ₂ ).-   Fact 4: Since the points P ε    ₁ and φ(P) are linearly independent they generate a group isomorphic    to    _(q)×    _(q). We denote this group of points by E[q].    Let    ₂ be the subgroup of    *_(p) ₂ of order q. The Weil pairing on the curve E(    _(p) ₂ ) is a mapping e: E[q]×E[q] →    ₂. (This map is defined and discussed in the section below entitled    Description of the Weil Pairing.) For any Q, R ε E(    _(p)) the Weil pairing satisfies e(Q, R)=1. In other words, the Weil    pairing is degenerate on E(    _(p)), and hence degenerate on the group    ₁. To get a non-degenerate map we define the modified Weil pairing    ê:    ₁×    ₁→    ₂ as follows:

{circumflex over (e)}(P,Q)=e(P,φ(Q))

The modified Weil pairing satisfies the following properties:

-   1. Bilinear: For all P, Q ε    ₁ and for all a, b ε    we have ê(aP, bQ)=ê(P, Q)^(ab).-   2. Non-degenerate: If P is a generator of    ₁ then ê(P, P) ε    *_(p) ₂ is a generator of    ₂.-   3. Computable: Given P, Q ε    ₁ there is an efficient algorithm to compute ê(P, Q) ε    ₂. (This Algorithm is Described in the Section Below Entitled    Description of the Weil Pairing.) Its running time is comparable to    exponentiation in    _(p).    Although the Computational Diffie-Hellman problem (CDH) appears to    be hard in the group    _(i), the Decision Diffie-Hellman problem (DDH) is easy in    ₁.    BDH Parameter Generator    ₁: Given a security parameter 2<k ε    the BDH parameter generator picks a random k-bit prime q and finds    the smallest prime p such that (1) p=2 mod 3, (2) q divides p+1,    and (3) q² does not divide p+1. We write p=lq+1. The group    ₁ is the subgroup of order q of the group of points on the curve    y²=x³+1 over    _(p). The group    ₂ is the subgroup of order q of    *_(p) ₂ . The bilinear map ê:    ₁×    ₁→    ₂ is the modified Weil pairing defined above.

The BDH parameter generator

₁ is believed to satisfy the BDH assumption asymptotically. However,there is still the question of what values of p and q can be used inpractice to make the BDH problem sufficiently hard. It is desirable thatwe can ensure, at the very least, that the discrete log problem in

₁ is sufficiently hard. As pointed out earlier, the discrete log problemin

₁ is efficiently reducible to discrete log in

₂. Hence, computing discrete log in

*_(p) ₂ is sufficient for computing discrete log in

₁. In practice, for proper security of discrete log in

*_(p) ₂ it is desirable to use primes p that are at least 512-bits long(so that the group size is at least 1024-bits long). Consequently, insome embodiments, this BDH parameter generator is used with primes pthat may be 512-bits long or more.

An Admissible Encoding Function: MapToPoint

Let

₁,

₂ be two groups generated by

₁ as defined above. Recall that an IBE system discussed earlier uses ahash function H₁: {0, 1}*→

*₁. It suffices to have a hash function H₁: {0, 1}*→A for some set A,and an admissible encoding function L: A→

₁. In what follows the set A will be

_(p), and the admissible encoding function L will be called MapToPoint,which may be used in various embodiments of the present invention.

In this example, let p be a prime satisfying p=2 mod 3 and p=lq−1 forsome prime q>3. In this exemplary embodiment, q does not divide l (i.e.q² does not divide p+1). Let E be the elliptic curve y²=x³+1 over

_(p). Let

₁ be the subgroup of points on E of order q. In addition, a hashfunction H₁: {0, 1}*→

_(p) is provided.

In this exemplary embodiment, algorithm MapToPoint works as follows oninput y_(o) ε

_(p):

-   1. Compute x₀=−1)^(1/3)=(y₀ ²−1)^((2p-1)/3) ε    _(p).-   2. Let Q=(x₀, y₀) ε E(    _(p)) and set Q_(ID)=ε    ₁.-   3. Output MapToPoint(y₀)=Q_(ID).

This completes the description of MapToPoint.

We note that there are l−1 values of y₀ ε

_(p) for which lQ=l(x₀, y₀)=O (these are the non-O points of orderdividing l). Let B ⊂

_(p) be the set of these y₀. When H₁(ID) is one of these l−1 valuesQ_(ID) is the identity element of

₁. It is extremely unlikely for H₁(ID) to hit one of these points—theprobability is 1/q<½^(k). Hence, for simplicity we say that H₁(ID) onlyoutputs elements in

_(p\)B, i.e. H₁: {0, 1}*→

_(p)\B. In other embodiments, algorithm MapToPoint can be easilyextended to handle the values y₀ ε B by hashing ID multiple times usingdifferent hash functions.

Proposition 8 MapToPoint:

_(p)\B→

*₁ is an admissible encoding function.Proof The map is clearly computable and is a l-to-1 mapping. It remainsto show that L is samplable. Let P be a generator of E(

_(p)). Given a Q ε

*₁ the sampling algorithm

_(S) does the following: (1) pick a random b ε {0, . . . l−1}, (2)compute Q′=l⁻¹·Q+bqP=(x, y), and (3) output

_(S)(Q)=y ε

_(p). Here l⁻¹ is the inverse of l in

*_(q). This algorithm outputs a random element from the l elements inMapToPoint⁻¹ (Q) as required. □

A Detailed Example of an IBE System

Using the BDH parameter generator

₁ and the admissible encoding function MapToPoint we obtain thefollowing detailed example of an embodiment of an IBE system.

Setup: Given a security parameter k ε

⁺, the algorithm works as follows:

-   -   Step 1: Run        ₁ on input k to generate a k-bit prime q and a prime p=2 mod 3        such that q divides p+1. Let E be the elliptic curve defined by        y²=x³+1 over        _(p). Choose an arbitrary P ε E(        _(p)) of order q.    -   Step 2: Pick a random s ε        *_(q) and set P_(pub)=sP.    -   Step 3: Pick four hash functions: H₁: {0, 1}*→        _(p); H₂:        _(p) ₂ →{0, 1}^(n) for some n;        -   H₃: {0, 1}^(n)×{0, 1}^(n)→            *_(q), and a hash function H₄: {0, 1}^(n)→{0, 1}^(n).            The message space is            ={0, 1}^(n). The ciphertext space is            =E(            _(p))×{0, 1}^(n). The system parameters are params=            p, q, n, P, P_(pub), H₁, . . . H₄            . The master-key is s ε            *_(q).            Extract: For a given string ID ε {0, 1}* the algorithm            builds a private key d as follows:

Step 1: Compute MapToPoint(H₁(ID))=Q_(ID) ε E(

_(p)) of order q.

Step 2: Set the private key d_(ID) to be d_(ID)=sQ_(ID) where s is themaster key.

Encrypt: To encrypt M ε {0, 1}^(n) under the public key ID do thefollowing:

Step 1: Compute MapToPoint(H₁/(ID))=Q_(ID) ε E(

_(p)) of order q.

Step 2: Choose a random σ ε {0, 1}^(n).

Step 3: Set r=H₃(σ, M).

Step 4: Set the ciphertext to be

C=

rP,σ⊕H ₂(g _(ID) ^(r)),M⊕H ₄(σ)

where g _(ID) =ê(Q _(ID) ,P _(pub))ε

_(p) ₂

-   Decrypt: Let C=    U, V, W    ε    be a ciphertext encrypted using the public key ID. If U ε E(    _(p)) is not a point of order q reject the ciphertext. To decrypt C    using the private key d_(ID) do:

Step 1. Compute V⊕H₂(ê(d_(ID), U))=σ.

Step 2. Compute W⊕H₄(σ)=M.

Step 3. Set r=H₃(σ, M). Test that U=rP. If not, reject the ciphertext.

Step 4. Output M as the decryption of C.

Performance. In this embodiment, algorithms Setup and Extract are verysimple. At the heart of both algorithms is a standard multiplication onthe curve E(

_(p)). Algorithm Encrypt requires that the encryptor compute the Weilpairing of Q_(ID) and P_(pub). Note that this computation is independentof the message, and hence can be done once and for all. Once g_(ID) iscomputed the performance of this embodiment is almost identical tostandard ElGamal encryption. We also note that the ciphertext length ofthe exemplary embodiment of BasicIdent is the same as in regular ElGamalencryption in

_(p). Decryption is a simple Weil pairing computation.Security. The security of the detailed exemplary embodiment justdescribed follows directly from Theorem 4 and Theorem 7.Corollary 9 The detailed exemplary embodiment described above is achosen ciphertext secure IBE system (i.e. IND-ID-CCA in the randomoracle model) assuming the BDH parameter generator G₁ satisfies the BDHassumption.

Extensions and Observations Tate Pairing and Other Curves.

Embodiments of our IBE system work with efficiently computable bilinearmaps ê:

₁×

₁→

₂ between two groups

₁,

₂ where the BDH assumption holds. Many different elliptic curves maygive rise to such maps. For example, one could use the curve y²=x³+xover

_(p) with p=3 mod 4 and its endomorphism φ: (x, y)→(−x, iy) where i²=−1.

In an alternative embodiment, one may use a family of nonsupersingularelliptic curves over

_(p) discussed by Miyaji et al. (A. Miyaji, M. Nakabayashi, S. Takano,“New explicit condition of elliptic curve trace for FR-reduction”, IEICETrans. Fundamentals, Vol. E84 A, No. 5, May 2001). For example, to use acurve E/

_(p) in this family one can take

₁ to be a cyclic subgroup of E(

_(p) ₆ ) (that is not contained in E(

_(p))) and then use the trace map on the curve E as the endomorphism φused to define the pairing ê. We also note that both encryption anddecryption in FullIdent can be made faster in alternate embodiments byusing the Tate pairing on elliptic curves rather than the Weil pairing.In other embodiments, suitable bilinear maps may be derived from abelianvarieties.

Asymmetric Pairings

As mentioned earlier, embodiments of our IBE system are not limited tosymmetric maps, but may include asymmetric maps as well. In other words,embodiments generally may use maps of the form ê:

₀×

₁→

₂ where

₀,

₁ are groups of prime order q. When

₀ and

₁ are equal we say the map is symmetric. When

₀ and

₁ are not equal we say the map is asymmetric.

The elements Q_(ID) and P in the asymmetric case are members of

₀ and

₁, respectively (or vice versa), and the target group of the hashfunction H₁ is selected accordingly. However, to make the proof ofsecurity go through (Lemma 2 in particular) we use a slightly strangelooking complexity assumption which we call the co-BDH assumption: givenrandom P, aP, bP ε

₁ and Q, aQ, cQ ε

₀ no polynomial time algorithm can compute ê(P, Q)^(abc) withnon-negligible probability. If one is uses this assumption then forembodiments using a curve E/

_(p) from Miyaji et al. (as just described above) one can take

₁ to be a cyclic subgroup of E(

_(p)) of order q and

₀ to be a different cyclic subgroup of E(

_(p) ₆ ) of order q. This will result in a more efficient system thanthe method described in the preceding paragraph for using these curves.

Distributed PKG

In exemplary embodiments of an IBE system it is desirable that themaster-key stored at the PKG be protected. One way of protecting thiskey is by distributing it among different sites using techniques ofthreshold cryptography. Embodiments of our IBE system support this in avery efficient and robust way. Recall that in some embodiments discussedabove, the master-key may be some s ε

*_(q) and the PKG uses the group action to compute a private key from sand Q_(ID), where Q_(ID) is derived from the user's public key ID. Forexample, d_(ID)=sQ_(ID). A distributed PKG embodiment can be implementedin a t-out-of-n fashion by giving each of the n PKGs one share s_(i) ofa Shamir secret sharing of s mod q. Each of the n PKGs can use its shares_(i) of the master key to generate a corresponding share d_(i) of aprivate key d_(ID) by calculating d_(i)=s_(i)Q_(ID). The user can thenconstruct the entire private key d_(ID) by requesting from t of the nPKGs its share d_(i) of the private key, then combining the shares bycalculating d_(ID)=Σ_(i)λ_(i)d_(i), where the λ_(i)'s are theappropriate Lagrange interpolation coefficients.

Furthermore, it is easy to make this embodiment robust against dishonestPKGs using the fact that DDH is easy in

₁. During setup each of the n PKGs publishes P_(i)=s_(i)P. During a keygeneration request the user can verify that the response from the i'thPKG is valid by testing that:

{circumflex over (e)}(d _(i) ,P)={circumflex over (e)}(Q _(ID) ,P _(i))

Thus, a misbehaving PKG will be immediately caught. There is no need forzero-knowledge proofs as in regular robust threshold schemes. The PKG'smaster-key can be generated in a distributed fashion using thetechniques of R. Gennaro et al. (R. Gennaro, S. Jarecki, H. Krawczyk, T.Rabin, “Secure Distributed Key Generation for Discrete-Log BasedCryptosystems”, Advances in Cryptology—Enrocrypt '99, Lecture Notes inComputer Science, Vol. 1592, Springer-Verlag, pp. 295-310, 1999). Usingthis technique, the PKGs can execute a cooperative protocol that allowsthem to jointly generate their respective shares of the master keywithout the master key ever existing in any one place.

Note that a distributed master-key embodiment also enables thresholddecryption on a per-message basis, without any need to derive thecorresponding decryption key. For example, threshold decryption ofBasicIdent ciphertext (U, V) is straightforward if each PKG respondswith ê(s_(i)Q_(ID, U).)

FIG. 5 is a block diagram illustrating a distributed PKG system,according to an embodiment of the invention. FIG. 5 includes a sendersystem 501, receiver system 502 and three PKGs (PKG A 503, PKG B 504 andPKG C 505). In one embodiment illustrating a 2-out-of-3 sharing, each ofthree PKGs contains a different share of a master key, and any two ofthe three are able to derive the master key. As shown in the figure, PKGA 503, PKG B 504, and PKG C 505 include, respectively, master key shares₁, 511, master key share s₂, 512, and master key share s₃, 513. In2-out-of-3 sharing, any two out of these three PKGs could combine theirshares to determine the master key, although in this embodiment each PKGsecretly holds its master key share.

Sender system 501 sends a message to receiver 502. The message 514 maybe encrypted using a public key based on an identifier ID of thereceiver. In order to obtain the corresponding private key, the receiversystem queries two of the three PKGs using, for example, the receiver'sidentity or public key. As shown in the figure, receiver system 502makes queries 506 and 507 to PKG A 503 and PKG B 504, respectively, inorder to obtain two shares of the private key. In response to thequeries, PKG A 503 and PKG B 504 return, respectively, share d₁, 508,and share d₂, 509, of private key d, 510. Receiver system 502 is thenable to assemble the corresponding private key d_(ID), which correspondsto the public key with which the message 514 was encrypted. Moregenerally, the receiver could have selected to query any two of thethree PKGs. For example, receiver system 502 alternatively could havequeried PKGs B and C and combined private key shares d₂ and d₃ toproduce the private key 510. These techniques easily generalize toprovide similar embodiments using t-out-of n sharing.

Sender system 501, receiver system 502 as well as PKGs 503, 504 and 505may be each implemented as computer systems which include elements suchas processors and computer-readable media such as memory and otherstorage devices. Communication between the respective elements may takeplace using data packets sent over data networks, or any of variousother forms of electronic and data transmission and communication. Thecommunication may transpire over various architectures of communication,such as a computer network, such as the Internet, with various wired,wireless and other communications media.

Working in Subgroups

In an alternative embodiment of the detailed IBE system described above,performance may be improved by working in a comparatively small subgroupof the curve. For example, choose a 1024-bit prime p=2 mod 3 with p=aq−1for some 160-bit prime q. The point P is then chosen to be a point oforder q. Each public key ID is converted to a group point by hashing IDto a point Q on the curve and then multiplying the point by a. Thesystem is secure if the BDH assumption holds in the group generated byP. The advantage of this embodiment is that the Weil computation is doneon points of small order, and hence is much faster.

IBE Implies Signatures

Various IBE techniques described above can be used to provide public keysignature systems and methods. The intuition is as follows. The privatekey for the signature scheme is the master key for the IBE scheme. Thepublic key for the signature scheme is the set of global systemparameters for the IBE scheme. The signature on a message M is the IBEdecryption key for ID=M. To verify a signature, choose a random messageM′, encrypt M′ using the public key ID=M, and then attempt to decryptusing the given signature on M as the decryption key. If the IBE systemis IND-ID-CCA, then the signature scheme is existentially unforgeableagainst a chosen message attack. Note that, unlike most signatureschemes, this signature verification embodiment is randomized. Thisshows that the IBE techniques described herein may encompass both publickey encryption and digital signatures. Signature schemes derived fromthese approaches can be used to provide interesting properties, asdescribed by Boneh et al. (D. Boneh, B. Lynn, H. Shacham, “Shortsignatures from the Weil pairing”, in Advances in Cryptology—AsiaCrypt2001, Lecture Notes in Computer Science, Vol. 2248, Springer-Verlag, pp.514-532, 2001, which is incorporated herein by reference).

Escrow ElGamal Encryption

In this section we show that various IBE techniques described above canbe used to provide an ElGamal encryption system embodiment having globalescrow capability. In this embodiment, a single escrow key enables thedecryption of ciphertexts encrypted under any public key.

In one exemplary embodiment, the ElGamal escrow system works as follows.The Setup is similar to that for BasicIdent. Unlike the identity-basedBasicIdent, each user selects a secret random number and uses it togenerate a public/private key pair. A sender and receiver can then useEncrypt and Decrypt to communicate an encrypted message. The message issecure except for an escrow who can use a master key s to decrypt themessage.

FIG. 6 is a block diagram illustrating elements in a cryptosystem withescrow decryption capability according to an embodiment of theinvention. The system includes a sender system 601 with encryption logic610, receiver system 602 with decryption logic 611, escrow agent system604 and broadcasting system 605. Broadcast system 605 sends systemparameters to participants such as escrow agent system 604, receiversystem 602 and sender system 601. The receiver system 602 selects aprivate key x, 607, and uses it to generate a public key P_(pub)=xP,606, which is then published. The private key x and the public keyP_(pub) form a complementary key pair. Using the public key P_(pub)=xP,606, sender system 601 encrypts a message M with encryption logic 610.Sender system 601 sends a resulting encrypted message 603 to receiver602. Receiver system 602 decrypts the message with decryption logic 611using the private key x, 607. Escrow agent system 604 may interceptmessage 603 and, using the escrow agent key s, 609, public keyP_(pub)=xP, 606, and decrypt message 603 with decryption logic 612. Inan alternate embodiment, broadcast system 605 and escrow agent 604 maybe a single entity. In yet another embodiment, the escrow agent key smay be shared in a manner such as in the distributed PKG embodimentsdescribed earlier.

In more detail, an exemplary embodiment of the technique involves thefollowing procedures:

-   Setup: Let    be some BDH parameter generator. Given a security parameter k ε    ⁺, the algorithm works as follows:    -   Step 1: Run        on input k to generate a prime q, two groups        ₁,        ₂ of order q, and an admissible map ê:        ₁×        ₁→        ₂. Let P be some generator of        ₁.    -   Step 2: Pick a random s ε        *_(q) and set Q=sP.    -   Step 3: Choose a cryptographic hash function H:        ₂→{0, 1}^(n).    -   The message space is        ={0, 1}^(n). The ciphertext space is        =        ₁×{0, 1}^(n). The system parameters are params=        q,        ₁,        ₂, ê, n, P, Q, H        . The escrow key is s ε        *_(q).-   keygen: A user generates a public/private key pair for herself by    picking a random x ε    *_(q) and computing P_(pub)=xP ε    ₁. Her private key is x (or xQ), her public key is P_(pub).-   Encrypt: To encrypt M ε {0, 1}^(n) under the public key P_(pub) do    the following: (1) pick a random r ε    *_(q), and (2) set the ciphertext to be:

C=

rP, M⊕H(g ^(r))

where g={circumflex over (e)}(P _(pub) ,Q)ε

₂.

-   -   This encryption technique is also illustrated in FIG. 7, where        the sender obtains the system parameters and elements P and Q=sP        in block 700, and obtains the recipient's public key P_(pub)=xP        in block 710. The sender then selects a random r and computes a        message key in block 720. The message key is then used to        encrypt a message in block 730. The sender then transmits an        encapsulated key rP and encrypted message V to the receiver.

-   Decrypt: Let C=    U, V    be a ciphertext encrypted using P_(pub). Then U ε    ₁. To decrypt C using the private key x do:

V⊕H({circumflex over (e)}(U,xQ))=M.

-   -   As illustrated in FIG. 8, the receiver obtains the system        parameters and elements P and Q=sP in block 800, then obtains        the encrypted message V and encapsulated key rP from the sender        in block 810. The receiver then computes the message key in        block 820, and uses it to decrypt the message in block 830.    -   To see that the message keys computed by the sender and receiver        are the same, note that the sender knows the secret r as well as        the public Q=sP and P_(pub)=xP, and uses these to compute a key        from ê(sP, xP)^(r). The receiver, on the other hand, knows the        secret x as well as the public Q=sP and rP, and uses these to        compute a message key from ê(rP, x(sP)). The bilinearity of ê        implies that ê(sP, xP)^(r)=ê(rP, x(sP)), so the sender and        receiver compute the same message key.

-   Escrow-decrypt: The purpose of this embodiment is to permit escrow    decryption of otherwise secure communications. To decrypt C=    U, V    using the escrow key s, compute:

V⊕H({circumflex over (e)}(U,sP _(pub)))=M.

-   -   As shown in FIG. 9, the escrow obtains the system parameters and        element P in block 900, then in block 910 obtains the        recipient's public key xP, and obtains the encrypted message V        and encapsulated key rP from the sender. The escrow then        computes the message key in block 920, and uses it to decrypt        the message in block 930. The escrow can compute the message key        from the knowledge of s, rP, and xP.        A standard argument shows that assuming that BDH is hard for        groups generated by G the system of this embodiment has semantic        security in the random oracle model (recall that since DDH is        easy we cannot prove semantic security based on DDH). Yet, the        escrow agent can decrypt any ciphertext encrypted using any        user's public key. The decryption capability of the escrow agent        can be distributed using the PKG distribution techniques        described earlier.

Another embodiment uses a similar hardness assumption, with an ElGamalencryption system with non-global escrow. In this embodiment, each userconstructs a public key with two corresponding private keys, and givesone of the private keys to the trusted third party. The trusted thirdparty maintains a database of all private keys given to it by thevarious users. Although both private keys can be used to decrypt, onlythe user's private key can be used simultaneously as the signing key fora discrete logarithm based signature scheme.

Various other cryptographic systems can be devised based on theprinciples illustrated in the above embodiments. For example, threeentities A, B, and C can communicate securely as a group by privatelyselecting random integers a, b, c and publishing public keys aP, bP, cP.One of them, such as A, could encrypt a message using the message keyê(bP, cP)^(r) and transmit it with rP. Then B could decrypt the messageby calculating ê(cP, rP)^(b) and C could decrypt it by calculating ê(bP,rP)^(c). Similarly, B could send a message to A and C, or C could send amessage to A and B.

In another possible embodiment, two of the three entities, say A and B,could publish a joint public key abP. Then C could encrypt a messageusing the message key ê(abP, cP)^(r) and transmit it with rP. Thenneither A nor B alone could decrypt the message, but both A and Btogether could compute ê(cP, rP)^(ab) and jointly decrypt the message.This technique generalizes to any number of entities. For example, Ccould join A and B by using abP to compute and publish the three-wayjoint public key abcP. Then anyone could encrypt a message using themessage key ê(abcP, xP)^(r) and transmit it with rP. Then only A and Band C together could compute ê(xP, rP)^(abc) and jointly decrypt themessage.

Threshold Decryption.

Embodiments of the invention enable n entities to have shares of aprivate key d_(ID) corresponding to a given public key ID, so thatmessages encrypted using ID can only be decrypted if t of the n entitiescollaborate. The private key d_(ID) is never reconstructed in a singlelocation. Embodiments of our IBE system may support this as follows.

Recall that in other embodiments the private key d_(ID)=sQ_(ID) where sε

*_(q) is the master-key. Instead, let, s₁, . . . , s_(n) ε

*_(q) be a t-out-of-n Shamir secret sharing of the master-key s. Each ofthe n users is given d_(i)=s_(i)Q_(ID). To decrypt a ciphertext

U, V

encrypted using the key ID each user locally computes g_(i)=ê(U, d_(i))and sends g_(i) ε

₂ to the user managing the decryption process. That user then combinesthe decryption shares by computing g_(ID)=π_(i)g_(i) ^(λi) where λ_(i)are the appropriate Lagrange interpolation coefficients used in Shamirsecret sharing. The message is then obtained by computingH₂(g_(ID))⊕V=M.

Those skilled in the art of cryptography will be able to devise manyother schemes that employ the basic principles of the present invention.

Applications of Identity-Based Encryption

One application for embodiments of identity-based encryption is to helpthe deployment of a public key infrastructure. In this section, we showseveral embodiments of this and other applications.

Revocation of Public Keys

In this embodiment, the sender may encrypt using a public key derivedfrom a piece of information containing a time element, such as a year,date or other time, to help provide key expiration or other forms oftemporal key management. For example, in one embodiment, key expirationcan be done by having Alice encrypt e-mail sent to Bob using the publickey: “bob@company.com|current-year”. In doing so Bob can use his privatekey during the current year only. Once a year Bob needs to obtain a newprivate key from the PKG. Hence, we get the effect of annual private keyexpiration. Note that unlike the existing public key infrastructure,Alice does not need to obtain a new certificate from Bob every time Bobrefreshes his private key.

One may make this approach more granular in other embodiments byencrypting e-mail for Bob using “bob@company.com|current-date”, or usinganother time stamp. This forces Bob to obtain a new private key everyday. This embodiment may be used in a corporate context where the PKG ismaintained by the corporation. With this approach key revocation is verysimple: when Bob leaves the company and his key needs to be revoked, thecorporate PKG is instructed to stop issuing private keys for Bob'se-mail address. As a result, Bob can no longer read his email. Theinteresting property is that Alice does not need to communicate with anythird party certificate directory to obtain Bob's daily public key.Hence, embodiments of identity based encryption can provide a veryefficient mechanism for implementing ephemeral public keys. Also notethat this embodiment can be used to enable Alice to send messages intothe future: Bob will only be able to decrypt the e-mail on the datespecified by Alice.

Managing User Credentials

An embodiment of the invention enables the management of usercredentials using an IBE system. The message is encrypted with a stringcontaining a credential identifier. For example, suppose Alice encryptsmail to Bob using the public key: “bob@company.com|current-yearclearance=secret”. Then Bob will only be able to read the email if onthe specified date he has secret clearance. Consequently, it is veryeasy to grant and revoke user credentials using the PKG.

FIG. 10 is a block diagram illustrating a system for managingcredentials in an identity based encryption system according to anembodiment of the invention. The system includes sender system 1001,receiver system 1002 and PKG 1003. Each such system may be implementedas a computer system such as a client or server connected to a computernetwork. Accordingly, sender 1001, receiver 1002 and PKG 1003 may eachcontain processors, such as processor 1014, processor 1013 and processor1012. Additionally, these systems may include computer-readable storagemedia, such as computer memory, and may additionally include interfacesto a computer network, including technology allowing for communicationwith a wired, wireless or other network. Sender system 1001 may includea software plug-in 1017. Such a plug-in may comprise a software modulewhich performs cryptographic functions. The plug-in includes, accordingto an embodiment of the invention, items such as cryptographic logic1004. Plug-in 1017 may be distributed to various computers such assender system 1001 and receiver system 1002 through a network in orderto roll out functionality associated with identity-based encryption andother communication functionality. Parameters 1015 from a system such asPKG 1003 are also distributed over a computer network or othercommunications medium to senders and receivers, such as sender system1001 and receiver system 1002, who may then use them in conjunction withplug-in 1017 when encrypting or decrypting messages. In one embodiment,plug-in 1017 is distributed together with parameters 1014. In analternate embodiment, parameters 1015 may be distributed separately.

Sender system 1001 encrypts a message M using encryption logic 1004 inplug-in 1017. Encryption logic 1004 encrypts the message usingencryption key 1011, which is based on selected credential 1005 and anidentification 1016 of the intended receiver of the message. In someembodiments, the key may be based on other information as well. Thesender system 1001 sends the receiver system 1002 information 1006,e.g., in the form of a data packet transmitted over a network or othercommunication medium. The information 1006 sent to receiver system 1002contains the encrypted message and may also contain information 1007regarding the credential 1005 used as part of the basis for theencryption key.

Either before or after receiving information 1006, receiver system 1002sends a request 1009 to PKG 1003. In one embodiment, the request 1009may include the receiver's identity 1016 and may also includeinformation related to the selected credential 1005. In response, PKG1003 verifies the credential of receiver 1002 using credential checklogic 1008. Such logic may be implemented in software, hardware or acombination thereof. If the credential is verified as belonging to thereceiver, then PKG 1003 provides a response 1010 to receiver 1002, whichincludes a private decryption key 1018 corresponding to the encryptionkey 1011. Using the private decryption key, the receiver then maydecrypt the encrypted message contained in information 1006 to recoverthe original message M. Thus, by including a credential as part of anencryption key, embodiments such as this one allow a sender to encrypt amessage intended for a receiver, where the decryption of the message bythe receiver is contingent upon the validity of the receiver'scredential.

Delegation of Decryption Keys

Another application for embodiments of IBE systems is delegation ofdecryption capabilities. We give two exemplary embodiments, describedwith reference to a user Bob who plays the role of the PKG. Bob runs thesetup algorithm to generate his own IBE system parameters params and hisown master-key. Here we view params as Bob's public key. Bob obtains acertificate from a CA for his public key params. When Alice wishes tosend mail to Bob she first obtains Bob's public key params from Bob'spublic key certificate. Note that Bob is the only one who knows hismaster-key and hence there is no key-escrow with this setup.

-   1. Delegation to a laptop. Suppose Alice encrypts mail to Bob using    the current date as the IBE encryption key (she uses Bob's params as    the IBE system parameters). Since Bob has the master-key he can    extract the private key corresponding to this IBE encryption key and    then decrypt the message. Now, suppose Bob goes on a trip for seven    days. Normally, Bob would put his private key on his laptop. If the    laptop is stolen the private key is compromised. When using the IBE    system Bob could simply install on his laptop the seven private keys    corresponding to the seven days of the trip. If the laptop is    stolen, only the private keys for those seven days are compromised.    The master-key is unharmed. FIG. 11 is a block diagram illustrating    a system with key delegation according to an embodiment of the    invention. The system includes user system 1101 and target system    1102. The target system may comprise a computer such as a laptop    computer. User system 1101 includes a master key 1103, which is used    to generate decryption keys 1104. The decryption keys 1104 are    downloaded to the target system 1102. Using the techniques of key    revocation described above, these decryption keys may be valid only    for a limited time, thus providing additional security in the event    that target system 1101 is compromised. User system 1101 and target    system 1102 may include elements of computer systems such as memory    1106 and 1107 as well as processor 1105 and 1108. User system 1101    includes key generator logic 1109, which uses master key 1103 and    system parameters 1110 to generate private decryption keys 1104    based on information derived from a user ID 1113 and one or more    dates 1114 or other time stamps. Target system 1102 includes    decryption logic 1111, which uses the private decryption keys 1104    obtained from user system 1101 and system parameters 1110 to decrypt    an incoming encrypted message 1112. If message 1112 is encrypted    using public keys based on ID 1113 and one of the dates 1114, then    private decryption keys may be used to decrypt it. Thus the    decryption capabilities of target system 1102 may be limited to    messages associated with selected dates 1114. In an alternate    embodiment, the target system may be a data storage medium or    portable data storage device which can be connected as desired to    other computer systems, thereby enabling use of the decryption keys    on those systems.-   2. Delegation of duties. Suppose Alice encrypts mail to Bob using    the subject line as the IBE encryption key. Bob can decrypt mail    using his master-key. Now, suppose Bob has several assistants each    responsible for a different task (e.g. one is ‘purchasing’, another    is ‘human-resources’, etc.). In this embodiment, Bob may give one    private key to each of his assistants corresponding to the    assistant's responsibility. Each assistant can then decrypt messages    whose subject line falls within its responsibilities, but it cannot    decrypt messages intended for other assistants. Note that Alice only    obtains a single public key from Bob (params), and she uses that    public key to send mail with any subject line of her choice. The    mail can only be read by the assistant responsible for that subject.    More generally, embodiments of IBE can simplify various systems that    manage a large number of public keys. Rather than storing a big    database of public keys the system can either derive these public    keys from user names, or simply use the integers 1, . . . , n as    distinct public keys. For example, in a corporation, each employee    might have a unique employee number, and that number may serve also    as the employee's public key.

Return Receipt

FIG. 12 is a block diagram illustrating an encryption system with returnreceipt according to an embodiment of the invention. According to oneembodiment of the invention, a sender can receive an confirmation thatthe recipient has received an encrypted message. More generally, uponreceipt of a request for a decryption key from a receiver, the PKG takesan action separate from providing a decryption key to the receiver. Suchan action comprises providing an acknowledgement to the sender thatindicates that the message was received, according to one embodiment.

An embodiment of a system having return receipt capability isillustrated in FIG. 12. The system includes sender system 1201,recipient system 1202 and PKG system 1203. The sender system 1201,receiver system 1202 and PKG system 1203 may be implemented as computersystems coupled to a computer network. For example, PKG 1203, sendersystem 1201 and receiver system 1202 may include processor 1212,processor 1213 and processor and 1214, respectively. These computersystems may include elements such as computer readable storage media,computer memory and other storage devices. Additionally, these systemsmay include interfaces to a computer network, including technologyallowing for communication from a wired, wireless or other network.Further, according an embodiment of the invention, communication betweenthe respective elements may take place using data packets sent over acomputer network, or using any of various other forms of electronic anddata transmission and communication.

The sender 1201 encrypts a message M and sends the resulting ciphertextto receiver 1202 in a data package 1204 that also may include returnreceipt request information 1209. The return receipt request informationmay contain, for example, a return address and a message identifiercorresponding to the particular message 1204. The message M is encryptedby the sender using encryption logic 1211 and an encryption key 1215.Encryption key 1215 may be based on a receiver ID (such as an e-mailaddress) 1216 and the return receipt request information 1209. Becausethe receiver ID and return receipt request information 1209 are used bythe sender to determine the encryption key 1215, the receiver 1202 needsa corresponding decryption key that can be used to decrypt the message.Accordingly, recipient system 1202, in response to receiving message1204, sends PKG 1203 a request 1206, which includes the return receiptrequest information 1209 and the receiver's ID, 1216. In response, PKG1203 sends to receiver 1202 the private decryption key 1205, whichreceiver then uses with decryption logic 1217 to decrypt the ciphertextof message 1204 and recover the original message M. In addition tosending receiver 1202 the decryption key 1205, PKG 1203 also sends areturn receipt 1207 to sender 1201. PKG 1203 may alternatively store thereceipt on storage media as part of a log rather than send a returnreceipt. Return receipt 1207 may include information such as the messageidentifier. Thus, sender 1201 receives proof that recipient 1202 hasreceived the message 1204. The system may be initialized by placingplug-in software in various systems, such as sender system 1201 andreceiver system 1202. Such plug-in software may include systemparameters, some of which may be derived from a system master key. Suchparameters, stored in local devices such as sender 1201 and receiver1202 are then used to generate encryption keys, perform encryption,perform decryption, and other functions, as appropriate.

Description of the Weil Pairing

In this section we describe the Weil pairing on elliptic curves and thenshow how to efficiently compute it using an algorithm. To be concrete wepresent an example using supersingular elliptic curves defined over aprime field

_(p) with p>3 (the curve y²=x³+1 over

with p=2 mod 3 is an example of such a curve). The following discussioneasily generalizes to computing the Weil pairing over other ellipticcurves.

Elliptic Curves and the Weil Pairing

We state a few elementary facts about supersingular elliptic curvesdefined over a prime field

_(p) with p>3:

-   Fact 1: A supersingular curve E/    _(p) (with p>3) contains p+1 points in    _(p). We let O denote the point at infinity. The group of points    over    _(p) forms a cyclic group of order p+1. For simplicity, let P be a    generator of this group and set n=p+1.-   Fact 2: The group of points E(    _(p) ₂ ) contains a point Q of order n which is linearly independent    of the points in E(    _(p)). Hence, E(    _(p) ₂ ) contains a subgroup which is isomorphic to the group    _(n) ². The group is generated by P ε E(    _(p)) and Q ε E(    _(p) ₂ ). We denote this group by E[p+1]=E[n].    We will be working with the Weil pairing e which maps pairs of    points in E[n] into    *_(p) ₂ , i.e. e: E[n]×E[n]→    *_(p) ₂ . To describe the pairing, we review the following concepts:-   Divisors A divisor is a formal sum of points on the curve E(    _(p) ₂ ). We write divisors as    =Σ_(P)a_(p)(P) where a_(P) ε    and P ε E(    _(p) ₂ ). For example,    =3(P₁)−2(P₂)−(P₃) is a divisor. We will only consider divisors    =Σ_(P)a_(p)(P) where Σ_(P)a_(p)=0.-   Functions Roughly speaking, a function ƒ on the curve E(    _(p) ₂ ) can be viewed as a rational function ƒ(x, y) ε    _(p) ₂ (x, y). For any point P=(x, y) ε E(    _(p) ₂ ) we define ƒ(P)=ƒ(x, y).-   Divisors of functions Let ƒ be a function on the curve E(    _(p) ₂ ). We define its divisor, denoted by (ƒ), as (ƒ)=Σ_(P)    ord_(P)(ƒ)·P. Here ord_(P)(ƒ) is the order of the zero that ƒ has at    the point P. For example, let ax+by+c=0 be the line passing through    the points P₁, P₂ ε E(    _(p) ₂ ) with P₁≠P₂. This line intersects the curve at third point    P₃ ε E(    _(p) ₂ ). Then the function ƒ(x, y)=ax+by+c has three zeroes P₁, P₂,    P₃ and a pole of order 3 at infinity. The divisor of ƒ is    (ƒ)=(P₁)+(P₂)+(P₃)−3(O).-   Principal divisors Let    be a divisor. If there exists a function ƒ such that (ƒ)=    then we say that    is a principal divisor. We know that a divisor    =Σ_(P)a_(p)(P) is principal if and only if Σ_(P)a_(p)=0 and    Σ_(P)a_(p)P=O. Note that the second summation is using the group    action on the curve. Furthermore, given a principal divisor    there exists a unique function ƒ (up to constant multiples) such    that (A)=(ƒ).-   Equivalence of divisors We say that two divisors    ,    are equivalent if their difference    −    is a principal divisor. We know that any divisor    =Σ_(P)a_(p)(P) (with Σ_(P)a_(P)=0) is equivalent to a divisor of the    form    ′=(Q)−(O) for some Q ε E. Observe that Q=Σ_(P)a_(P)P.-   Notation Given a function ƒ and a divisor    =Σ_(P)a_(p)(P) we define ƒ (    ) as ƒ(    )=π_(P)ƒ(P)^(aP). Note that since Σ_(P)a_(P)=0 we have that ƒ(    ) remains unchanged if instead of ƒ we use cf for any c ε    _(p) ₂ .    We are now ready to describe the Weil pairing of two points P, Q ε    E[n]. Let    _(P) be some divisor equivalent to the divisor (P)−(O). We know that    n    _(P) is a principal divisor (it is equivalent to n(P)−n(O) which is    clearly a principal divisor). Hence, there exists a function ƒ_(P)    such that (ƒ_(P))=n    _(P). Define    _(Q) and ƒ_(Q) analogously. The Weil pairing of P and Q is given by:

e  ( P , Q ) = f P  ( Q ) f Q  ( p )

This ratio provides the value of the Weil pairing of P and Q whenever itis well defined (i.e., whenever no division by zero has occurred). Ifthis ratio is undefined we use different divisors

_(P),

_(Q) to define e(P, Q). When P, Q ε E(

_(p) ₂ ) we have that e(P, Q) ε

_(p) ₂ .

We briefly show that the Weil pairing is well defined. That is, thevalue of e(P, Q) is independent of the choice of the divisor

_(P) as long as

_(P) is equivalent to (P)−(O) and

_(P) leads to a well defined value. The same holds for

_(Q). Let

_(P) be a divisor equivalent to

_(P) and let {circumflex over (ƒ)}_(p) be a function so that({circumflex over (ƒ)}_(P))=n

_(P). Then

_(P)=

_(P)+(g) for some function g and {circumflex over (ƒ)}_(P)=ƒ_(P)·g_(n).We have that:

e  ( P , Q ) = f ^ P  ( Q ) f Q  ( ^ P ) = f P  ( Q )  g  ( Q ) nf Q  ( P )  f Q  ( ( g ) ) = f P  ( Q ) f Q  ( P ) · g  ( n   Q) f Q  ( ( g ) ) = f P  ( Q ) f Q  ( P ) · g  ( ( f Q ) ) f Q  ( (g ) ) = f P  ( Q ) f Q  ( P )

The last equality follows from the following fact known as Weilreciprocity: for any two functions ƒ, g we have that ƒ ((g))=g((ƒ)).Hence, the Weil pairing is well defined.Fact 10 The Weil pairing has the following properties:

-   -   For all P ε E[n] we have: e(P, P)=1.    -   Bilinear: e(P₁+P₂, Q)=e(P₁, Q)·e(P₂, Q) and e(P, Q₁+Q₂)=e(P,        Q₁)·e(P, (Q₂).    -   When P, Q ε E[n] are collinear then e(P, Q)=1. Similarly, e(P,        Q)=e(Q, P)⁻¹.    -   n'th root: for all P, Q ε E[n] we have e(P, Q)^(n)=1.    -   Non-degenerate: if P satisfies e(P, Q)=1 for all Q ε E[n] then        P=O.

As discussed earlier, our detailed example of an embodiment of an IBEscheme uses the modified Weil pairing ê(P, Q)=e(P, φ(Q)), where φ is anautomorphism on the group of points of E.

Tate pairing. The Tate pairing is another bilinear pairing that has therequired properties for embodiments of our system. In variousembodiments, we slightly modify the original definition of the Tatepairing to fit our purpose. Define the Tate pairing of two points P, Q εE[n] as T(P, Q)=ƒ_(P)

where ƒ_(P) and

_(Q) are defined as earlier. This definition gives a computable bilinearpairing T: E[n]×E[n]→

₂.

Computing the Weil Pairing

Given two points P, Q ε E[n] we show how to compute e(P, Q) ε

*_(p) ₂ using O(log p) arithmetic operations in

_(p). We assume P≠Q. We proceed as follows: pick two random points R₁,R₂ ε E[n]. Consider the divisors

_(P)=(P+R₁)−(R₁) and

_(Q)=(Q+R₂)−(R₂). These divisors are equivalent to (P)−(O) and (Q)−(O)respectively. Hence, we can use

_(P) and

_(Q) to compute the Weil pairing as:

e  ( P , Q ) = f P  ( Q ) f Q  ( P ) = f P  ( Q + R 2 )  f Q  ( R1 ) f P  ( R 2 )  f Q  ( P + R 1 )

This expression is well defined with very high probability over thechoice of R₁, R₂ (the probability of failure is at most

$\left. {O\left( \frac{\log \mspace{14mu} p}{p} \right)} \right).$

In the rare event that a division by zero occurs during the computationof e(P, Q) we simply pick new random points R₁, R₂ and repeat theprocess.

To evaluate e(P, Q) it suffices to show how to evaluate the functionƒ_(P) at

_(Q). Evaluating ƒ_(Q)(

_(P)) is done analogously. We evaluate ƒ_(P)(

_(Q)) using repeated doubling. For a positive integer b define thedivisor

_(b) =b(P+R ₁)−b(R ₁)−(bP)+(O)

It is a principal divisor and therefore there exists a function ƒ_(b)such that (ƒ_(b))=

_(b). Observe that (ƒ_(P))=(ƒ_(n)) and hence, ƒ_(P)(

_(Q))=ƒ_(n)(

_(Q)). It suffices to show how to evaluate ƒ_(n)(

_(Q)).Lemma 11 There is an algorithm

that given ƒ_(b)(

_(Q)), ƒ_(c)(

_(Q)) and bP, cP, (b+c)P for some b, c>0 outputs ƒ_(b+c)(

_(Q)). The algorithm only uses a (small) constant number of arithmeticoperations in

_(p) ₂ .Proof We first define two auxiliary linear functions g₁, g₂:

-   1. Let a₁x+b₁y+c₁=0 be the line passing through the points bP and cP    (if b=c then let a₁x+b₁y+c₁=0 be the line tangent to E at bP).    Define g₁(x, y)=a₁x+b₁y+c₁.-   2. Let x+c₂=0 be the vertical line passing through the point (b+c)P.    Define g₂(x, y)=x+c₂    The divisors of these functions are:

(g ₁)=(bP)+(cP)+(−(b+c)P)−3(O)

(g ₂)=((b+c)P)+(−(b+c)P)−2(O)

By definition we have that:

_(b) =b(P+R ₁)−b(R ₁)−(bP)+(O)

_(c) =c(P+R ₁)−c(R ₁)+(cP)+(O)

_(b+c)=(b+c)(P+R ₁)−(b+c)(R ₁)−((b+c)P)+(O)

It now follows that:

_(b+c)=

_(b)+

_(c)+(g₁)−(g₂). Hence:

f b + c  ( Q ) = f b  ( Q ) · f c  ( Q ) · g 1  ( Q ) g 2  ( Q ) (1 )

This shows that to evaluate ƒ_(b+c)(

_(Q)) it suffices to evaluate g_(i)(

_(Q)) for all i=1, 2 and plug the results into equation 1. Hence, givenƒ_(b)(

_(Q)), ƒ_(c)(

_(Q)) and bP, cP, (b+c)P one can compute ƒ_(b+c)(

_(Q)) using a constant number of arithmetic operations. □

Denote the output of Algorithm

of Lemma 11 by

(ƒ_(b)(

_(Q)), ƒ_(c)(

_(Q)), bP, cP, (b+c)P)=ƒ_(b+c)(

_(Q)). Then one can compute ƒ_(P)(

_(Q))=ƒ_(c)(

_(Q)) using the following standard repeated doubling procedure. Letn=b_(m)b_(m−1) . . . b₁b₀ be the binary representation of n, i.e.n=Σ_(i=0) ^(m) b_(i)2^(i).

Init: Set Z=O, V=ƒ₀(

_(Q))=1, and k=0.Iterate: For i=m, m−1, . . . , 1, 0 do:

1: If b_(i)=1 then do: Set V=

(V, ƒ₁(

_(Q)), Z, P, Z+P), set Z=Z+P, and set k=k+1.

2: If i>0 set V=

(V, V, Z, Z, 2Z), set Z=2Z, and set k=2k.

3: Observe that at the end of each iteration we have z=kP and V=ƒ_(k)(

_(Q)).

Output: After the last iteration we have k=n and therefore V=ƒ_(n)(

_(Q)) as required.

To evaluate the Weil pairing e(P, Q) we run the above algorithm once tocompute ƒ_(p)(

_(Q)) and once to compute ƒ_(Q)(

_(p)). Note that the repeated squaring algorithm needs to evaluate ƒ₁(

_(Q)). This is easily done since the function ƒ₁(x, y) (whose divisor is(ƒ₁)=(P+R₁)−(R₁)−(P)+(O)) can be written out explicitly as follows:

-   -   1. Let a₁x+b₁y+c₁=0 be the line passing through the points P and        R₁. Define the function: g₁(x, y)=a₁x+b₁y+c₁.    -   2. Let x+c₂=0 be the vertical line passing through the point        P+R₁. Define the function: g₂(x, y)=x+c₂.    -   3. The function ƒ₁(x, y) is simply ƒ₁(x, y)=g₂(x, y)/g₁(x, y)        which is easy to evaluate in        _(p) ₂ .

1. A method, comprising: with computing equipment, encrypting data usinga public key that is formed using a time element, wherein encrypting thedata comprises encrypting the data using cryptographic system parametersand wherein encrypting the data comprises encrypting the data using abilinear map.
 2. The method defined in claim 1 further comprising:decrypting the encrypted data using a private key that corresponds tothe public key.
 3. The method defined in claim 2 further comprisingforming the public key using the time element, wherein the time elementincludes a date and wherein forming the public key comprises using thedate in forming the public key.
 4. The method defined in claim 1 furthercomprising forming the public key using the time element, wherein thetime element includes a future date and wherein forming the public keycomprises forming the public key using the future date.
 5. The methoddefined in claim 1 further comprising: obtaining a private key from aprivate key generator that corresponds to the public key; and decryptingthe encrypted data using the private key.
 6. A method, comprising: withcomputing equipment, encrypting data using a public key that is formedusing a credential associated with a user, wherein encrypting the datacomprises encrypting the data using cryptographic system parameters andwherein encrypting the data comprises encrypting the data using abilinear map.
 7. The method defined in claim 6 further comprising:decrypting the encrypted data using a private key that corresponds tothe public key.
 8. The method defined in claim 6 wherein the user hasidentifying information, the method further comprising forming thepublic key using the credential and using the identifying information ofthe user.
 9. The method defined in claim 6 further comprising: obtaininga private key that corresponds to the public key from a private keygenerator; and decrypting the encrypted data using the private key. 10.The method defined in claim 9 further comprising: at the private keygenerator, determining whether the user has the credential; and if theprivate key generator determines that the receiver has the credential,providing the private key over a communications network.
 11. The methoddefined in claim 6 further comprising: determining whether the user hasthe credential; and if it is determined that the user has thecredential, providing the private key.
 12. The method defined in claim 6wherein the user has an identity, the method further comprising: at aprivate key generator, receiving a request for the private key from theuser that includes information related to the credential and thatincludes the identity of the user; at the private key generator,verifying that the credential belongs to the user; and if the credentialis verified as belonging to the user, providing the requested privatekey to the user from the private key generator.
 13. The method definedin claim 6 wherein the user has an identity, the method furthercomprising: receiving a request for the private key from the user thatincludes information related to the credential and that includes theidentity of the user; in response to receiving the request, verifyingthat the credential belongs to the user; and if the credential isverified as belonging to the user, providing the requested private keyto the user.
 14. The method defined in claim 6 wherein the credentialincludes a security clearance, the method further comprising forming thepublic key using the security clearance.
 15. The method defined in claim6 wherein the user has identifying information, the method furthercomprising forming the public key using the identifying information ofthe user and a time element.
 16. The method defined in claim 6 furthercomprising: at a private key generator, determining whether the user hasthe credential; and if the private key generator determines that theuser has the credential, providing the private key over a communicationsnetwork.
 17. A method comprising: with computing equipment, performingcryptographic operations using a public key that is based on acredential associated with a party, wherein performing the cryptographicoperations comprises using a bilinear map.
 18. The method defined inclaim 17 further comprising: performing cryptographic operations using aprivate key corresponding to the public key.
 19. The method defined inclaim 18 further comprising: forming the public key using the credentialof the party.
 20. The method defined in claim 19 further comprising:obtaining a private key corresponding to the public key from a privatekey generator over a communications network.